An isometric topology optimization method and device based on thermoelastic metamaterial design

Abstract

The application provides an isometric topology optimization method and device based on thermal elastic metamaterial design, and relates to the field of isometric topology optimization. The method comprises the following steps: constructing a multi-material interpolation model corresponding to the thermal elastic metamaterial, and obtaining a plurality of groups of density distributions corresponding to a plurality of solid materials in the thermal elastic metamaterial design domain according to the multi-material interpolation model, one solid material corresponding to one density distribution; obtaining the relationship interpolation between the plurality of groups of density distributions, and obtaining the elastic modulus and the thermal expansion coefficient according to the relationship interpolation; calculating the equivalent elastic matrix corresponding to the thermal elastic metamaterial according to the elastic modulus, and calculating the thermal expansion coefficient matrix corresponding to the thermal elastic metamaterial according to the thermal expansion coefficient; performing discrete form processing on the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix through the isometric grid, constructing a constraint function according to the equivalent elastic matrix, constructing an objective function according to the thermal expansion coefficient matrix, and calculating the sensitivity information corresponding to the first design variable through the objective function and the constraint function; updating the first design variable according to the sensitivity information until the thermal expansion coefficient meets the convergence condition, and obtaining the second design variable when the convergence condition is met through the isometric topology design model of the negative thermal expansion metamaterial; and optimizing the thermal elastic metamaterial according to the second design variable. The application solves the problem that the finite element method is difficult to guarantee the efficient and high-precision design of the thermal elastic metamaterial when dealing with complex geometric structures.

Description

Technical Field

[0001] The present application relates to the field of isogeometric topology optimization, and in particular to an isogeometric topology optimization method and device based on thermoelastic metamaterial design. Background Art

[0002] Currently, the design methods of thermoelastic metamaterials with extreme thermal expansion coefficients mostly rely on the designer's intuition and experience of the unit cells that make up the metamaterial. Topology optimization uses optimization algorithms to find the optimal topological layout within the unit cell, which can greatly increase the design freedom and customize the design of metamaterials with different shapes while meeting certain constraints. Therefore, it is very necessary to consider the topological optimization design scheme of thermoelastic metamaterials.

[0003] Traditional topology optimization design methods primarily employ the finite element method (FEM) for numerical analysis. In this method, the finite element mesh only approximates the geometry of the design domain, rather than accurately describing it. The responses between adjacent finite elements only satisfy C0 continuity, meaning that the displacement field may be discontinuous or non-smooth at element boundaries. Because FEM meshes are typically polygonal or polyhedral, this geometric approximation leads to decreased computational accuracy. This makes it difficult to achieve efficient and high-precision design of thermoelastic metamaterials, especially when dealing with complex geometries.

[0004] Therefore, there is an urgent need for an isogeometric topology optimization method and device based on thermoelastic metamaterial design. Summary of the Invention

[0005] The present application provides a method and apparatus for isogeometric topology optimization based on thermoelastic metamaterial design, which solves the problem that the finite element method is difficult to ensure efficient and high-precision design of thermoelastic metamaterials when processing complex geometric structures.

[0006] In a first aspect of the present application, a method for isogeometric topology optimization based on thermoelastic metamaterial design is provided, the method comprising: constructing a multi-material interpolation model corresponding to the thermoelastic metamaterial, and obtaining multiple density distributions corresponding to multiple solid materials in the thermoelastic metamaterial design domain based on the multi-material interpolation model, and one density distribution corresponding to each solid material; obtaining a relationship interpolation between the multiple density distributions, and obtaining an elastic modulus and a thermal expansion coefficient based on the relationship interpolation; calculating an equivalent elastic matrix corresponding to the thermoelastic metamaterial based on the elastic modulus, and calculating a thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial based on the thermal expansion coefficient; discretizing the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix using an isogeometric grid, constructing a constraint function based on the equivalent elastic matrix, constructing an objective function based on the thermal expansion coefficient matrix, and calculating sensitivity information corresponding to a first design variable using the objective function and the constraint function; the first design variable being the multiple density distributions; updating the first design variable based on the sensitivity information until the thermal expansion coefficient meets a convergence condition, and obtaining a second design variable when the convergence condition is met through an isogeometric topology design model of a negative thermal expansion metamaterial; and optimizing the thermoelastic metamaterial based on the second design variable.

[0007] Optionally, constructing a multi-material interpolation model corresponding to the thermoelastic metamaterial specifically includes: dividing the thermoelastic metamaterial into units to obtain a plurality of units; obtaining the Young's modulus corresponding to each unit and obtaining the density distribution corresponding to each unit; and constructing the multi-material interpolation model according to the Young's modulus corresponding to each unit and the density distribution corresponding to each unit using the following formula:

[0008]

[0009] Among them, E ei is the Young's modulus at the i-th unit after interpolation, α ei is the thermal expansion coefficient of the i-th unit after interpolation, is the total density of the first solid material and the second solid material at unit i, and And when When , unit i is completely filled with solid phase, When , the unit i is completely filled with the void phase. is the type of solid at unit i, and when When , only the first solid material exists in unit i, and when When , only the second solid material exists in unit i, N is the number of units, where i ranges from 1 to N, p is the penalty factor, E1 is the Young's modulus corresponding to the first solid material, E2 is the Young's modulus corresponding to the second solid material, α1 is the thermal expansion coefficient corresponding to the first solid material, and α2 is the thermal expansion coefficient corresponding to the second solid material.

[0010] Optionally, a preset basis function is constructed and the density at the control point is obtained; based on the preset basis function and the density at the control point, the total density and the type of solid are calculated using the following formula:

[0011]

[0012] in, for The density at the corresponding control point, for, for The density at the corresponding control point, is, ic is the geometric center of the i-th unit, c i is the set of control points contained in the i-th unit, c ij c i The j-th control point in N ij is the preset basis function.

[0013] Optionally, constructing a preset basis function specifically includes: constructing a target node vector based on the order of the preset basis function and the number of control points; and constructing the preset basis function according to the target node vector.

[0014] Optionally, calculating the equivalent elastic matrix corresponding to the thermoelastic metamaterial according to the elastic modulus and calculating the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial according to the thermal expansion coefficient specifically includes: calculating the equivalent elastic matrix corresponding to the thermoelastic metamaterial using the following formula:

[0015]

[0016] in, is the equivalent elastic matrix, V is the volume fraction of the solid material, is the global unit strain field, ε rs ,ε rs is the local strain field, D pqrs is the local variable stiffness tensor, ij and kl represent the two macroscopic dimensions, and pq and rs represent the two microscopic dimensions. The thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial is calculated using the following formula: in, is the thermal expansion coefficient matrix, α rs is the thermal expansion coefficient at the microscopic scale.

[0017] Optionally, the sensitivity information corresponding to the first design variable is calculated through the objective function and the constraint function, specifically including: obtaining the first partial derivative of the objective function with respect to the first design variable, obtaining the second partial derivative of the constraint function with respect to the first design variable; and calculating the sensitivity information corresponding to the first design variable based on the first partial derivative and the second partial derivative.

[0018] Optionally, constructing a geometric topology design model of a negative thermal expansion metamaterial specifically includes: constructing a geometric topology design model of a negative thermal expansion metamaterial according to the following formula:

[0019]

[0020] Among them, ρ (1) is the total density of the first solid material and the second solid material at the control point, ρ (2) is the density of the second solid material at the control point, accounting for the proportion of the total density at the control point, M is, f(ρ (1) ,ρ (2) ) is, H is, α is the thermal expansion coefficient, for, where K is the stiffness matrix, U is the displacement of the control point, F is the external load applied at the control point, V1 is the volume fraction corresponding to the first solid material, and its upper bound is The lower bound is V2 is the volume fraction corresponding to the second solid material, and its upper bound is The lower bound is K bulk is the equivalent bulk modulus of the microstructure of the negative thermal expansion metamaterial, K set is the constrained minimum value of the equivalent bulk modulus.

[0021] In a second aspect of the present application, an isogeometric topology optimization device based on thermoelastic metamaterial design is provided, the device comprising a multi-material interpolation model construction module and a thermoelastic metamaterial design module, wherein:

[0022] The multi-material interpolation model construction module is used to construct a multi-material interpolation model corresponding to the thermoelastic metamaterial, and obtain multiple groups of density distributions corresponding to multiple solid materials in the thermoelastic metamaterial design domain and one density distribution corresponding to one solid material based on the multi-material interpolation model.

[0023] A thermoelastic metamaterial design module is used to obtain the relationship interpolation between multiple groups of density distributions, and obtain the elastic modulus and thermal expansion coefficient based on the relationship interpolation; calculate the equivalent elastic matrix corresponding to the thermoelastic metamaterial based on the elastic modulus, and calculate the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial based on the thermal expansion coefficient; discretize the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix respectively through equal geometric grids, and construct a constraint function based on the equivalent elastic matrix, and construct an objective function based on the thermal expansion coefficient matrix, and calculate the sensitivity information corresponding to the first design variable through the objective function and the constraint function; the first design variable is multiple groups of density distributions; the first design variable is updated according to the sensitivity information until the thermal expansion coefficient meets the convergence condition, and the second design variable when the convergence condition is met is obtained through a geometric topology design model such as a negative thermal expansion metamaterial; the thermoelastic metamaterial is optimized according to the second design variable.

[0024] In the third aspect of the present application, an electronic device is provided, including a processor, a memory, a user interface and a network interface, the memory is used to store instructions, the user interface and the network interface are used to communicate with other devices, and the processor is used to execute the instructions stored in the memory so that the electronic device performs any of the methods described above.

[0025] In a fourth aspect of the present application, a computer-readable storage medium is provided, wherein the computer-readable storage medium stores a computer program, and the computer program is executed by a processor to perform any of the above methods.

[0026] One or more technical solutions provided in the embodiments of this application have at least the following technical effects or advantages:

[0027] 1. Construct a multi-material interpolation model to obtain the density distribution of multiple solid materials in the design domain, and calculate the elastic modulus and thermal expansion coefficient of the material through interpolation. Then, based on the elastic modulus and thermal expansion coefficient, the equivalent elastic matrix and thermal expansion coefficient matrix of the thermoelastic metamaterial are calculated respectively. The objective function and constraint function are constructed based on this. The design variables are updated in combination with sensitivity analysis until the thermal expansion coefficient meets the convergence conditions, thereby obtaining the optimization results of the negative thermal expansion metamaterial that meets the design requirements. The multi-material interpolation model can accurately represent the complex material distribution and improve the design accuracy. At the same time, combined with the isogeometric topology optimization design, it can not only ensure the consistency of the geometric and physical models, but also flexibly adjust the design under the constraints, realizing the efficient and high-precision optimization design of negative thermal expansion or zero thermal expansion metamaterials.

[0028] 2. Segment the thermoelastic metamaterial into units, obtain the Young's modulus and density distribution of each unit, and build a multi-material interpolation model based on this. By accurately obtaining the Young's modulus and density distribution of the thermoelastic metamaterial, the multiphase properties of the thermoelastic metamaterial can be better described, improving the accuracy and flexibility of the thermoelastic metamaterial design.

[0029] 3. By constructing an isogeometric topological design model of negative thermal expansion metamaterials, and based on multiple constraints such as the material's density distribution, thermal expansion coefficient, and volume fraction, the elastic matrix and thermal expansion coefficient matrix are optimized to determine the optimal distribution of the material. This allows for accurate control of material distribution, optimization of the material's negative thermal expansion characteristics, and maximization of material performance under given constraints. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 1 is a flow chart of an isogeometric topology optimization method based on thermoelastic metamaterial design provided in an embodiment of the present application;

[0031] Figure 2 This is a schematic diagram of the relationship between unit density and control point density provided in an embodiment of the present application;

[0032] Figure 3 This is a schematic diagram of an initial random density distribution provided in an embodiment of the present application;

[0033] Figure 4 This is a schematic diagram of a method for applying periodic boundary conditions provided in an embodiment of the present application;

[0034] Figure 5 This is a schematic diagram of a global macroscopic strain field provided in an embodiment of the present application;

[0035] Figure 6 This is a schematic diagram of an optimized configuration of a thermoelastic metamaterial provided in an embodiment of the present application;

[0036] Figure 7 This is a schematic diagram of a 3×3 periodic arrangement configuration of a negative thermal expansion two-dimensional structure provided in an embodiment of the present application;

[0037] Figure 8 This is a schematic diagram of a module of an isogeometric topology optimization device based on thermoelastic metamaterial design provided in an embodiment of the present application;

[0038] Figure 9 This is a structural diagram of an electronic device provided in an embodiment of the present application.

[0039] Explanation of the accompanying drawings: 21, unit; 22, control point; 23, control point set; 51, first macroscopic global strain field; 52, second macroscopic global strain field; 53, third macroscopic global strain field; 54, global thermal strain field; 81, multi-material interpolation model construction module; 82, thermoelastic metamaterial design module; 901, processor; 902, communication bus; 903, user interface; 904, network interface; 905, memory. DETAILED DESCRIPTION

[0040] In order to enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions in the embodiments of this specification will be clearly and completely described below in conjunction with the drawings in the embodiments of this specification. Obviously, the described embodiments are only part of the embodiments of this application, not all of the embodiments.

[0041] The terms used in the following examples of the present application are only for the purpose of describing specific embodiments and are not intended to limit the present application. As used in the specification of the present application, the singular expressions "a", "an", "said", "above", "the", and "this" are intended to include plural expressions as well, unless the context clearly indicates otherwise. It should also be understood that the term "and / or" used in the present application refers to and includes any or all possible combinations of one or more of the listed items.

[0042] In the following, the terms "first" and "second" are used for descriptive purposes only and should not be understood to imply or suggest relative importance or implicitly indicate the number of the technical features indicated. Therefore, the features defined as "first" and "second" may explicitly or implicitly include one or more of the features. In the description of the embodiments of this application, unless otherwise specified, "plurality" means two or more.

[0043] In order to enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings.

[0044] Currently, the design methods of thermoelastic metamaterials with extreme thermal expansion coefficients mostly rely on the designer's intuition and experience of the unit cells that make up the metamaterial. Topology optimization uses optimization algorithms to find the optimal topological layout within the unit cell, which can greatly increase the design freedom and customize the design of metamaterials with different shapes while meeting certain constraints. Therefore, it is very necessary to consider the topological optimization design scheme of thermoelastic metamaterials.

[0045] Traditional topology optimization design methods primarily employ the finite element method (FEM) for numerical analysis. In this method, the finite element mesh only approximates the geometry of the design domain, rather than accurately describing it. The responses between adjacent finite elements only satisfy C0 continuity, meaning that the displacement field may be discontinuous or non-smooth at element boundaries. Because FEM meshes are typically polygonal or polyhedral, this geometric approximation leads to decreased computational accuracy. This makes it difficult to achieve efficient and high-precision design of thermoelastic metamaterials, especially when dealing with complex geometries.

[0046] Therefore, there is an urgent need for an isogeometric topology optimization method and device based on thermoelastic metamaterial design.

[0047] Please refer to Figure 1 , which shows a flow chart of an isogeometric topology optimization method based on thermoelastic metamaterial design provided in an embodiment of the present application, the flow chart mainly includes the following steps: S101 to S106.

[0048] Step S101 : constructing a multi-material interpolation model corresponding to the thermoelastic metamaterial, and obtaining multiple groups of density distributions corresponding to multiple solid materials in the thermoelastic metamaterial design domain according to the multi-material interpolation model, and one density distribution corresponding to one solid material.

[0049] Specifically, a multi-material interpolation model based on control point density is used to represent the density distribution of different materials within the thermoelastic microstructure design domain. Based on this multi-material interpolation model, multiple sets of density distributions corresponding to multiple solid materials in the thermoelastic metamaterial design domain are obtained, with one density distribution corresponding to each solid material. The multi-material interpolation model is constructed as follows: the thermoelastic metamaterial is segmented into units to obtain multiple units; the Young's modulus corresponding to each unit is obtained, and the density distribution corresponding to each unit is obtained; based on the Young's modulus corresponding to each unit and the density distribution corresponding to each unit, the multi-material interpolation model is constructed using the following formula:

[0050]

[0051] Among them, E ei is the Young's modulus at the i-th unit after interpolation, α ei is the thermal expansion coefficient of the i-th unit after interpolation, is the total density of the first solid material and the second solid material at unit i, and And when When , unit i is completely filled with solid phase, When , the unit i is completely filled with the void phase. is the type of solid at unit i, and when When , only the first solid material exists in unit i, and when When , unit i only has the second solid material, N is the number of units, where i ranges from 1 to N, p is the penalty factor, E1 is the Young's modulus corresponding to the first solid material, E2 is the Young's modulus corresponding to the second solid material, and E1 = 1, E2 = 1, α1 is the thermal expansion coefficient corresponding to the first solid material, and α2 is the thermal expansion coefficient corresponding to the second solid material. and The calculation is performed as follows: a preset basis function is constructed and the density at the control points is obtained; based on the preset basis function and the density at the control points, the total density and the type of solid are calculated using the following formula:

[0052]

[0053] in, for The density at the corresponding control point, for, for The density at the corresponding control point, is, ic is the geometric center of the i-th unit, c i is the set of control points contained in the i-th unit, c ij c i The j-th control point in N ij It is a preset basis function and a NURBS basis function. Please refer to Figure 2 , which shows a schematic diagram of the relationship between unit density and control point density provided by an embodiment of the present application, wherein unit 21 is the i-th unit, control point 22 is the control point corresponding to the i-th unit, and control point set 23 is set c i . N ij The construction method is as follows: construct a target node vector based on the order of the preset basis function and the number of control points; construct a preset basis function according to the target node vector, where the target node vector is Ξ={ξ1,ξ2,...,ξ m+q+1}, Ξ is the target node vector, {ξ1,ξ2,...,ξ m+q+1} is the node value in the target node vector, q is the order of the spline curve, and m is the number of control points, then:

[0054]

[0055] Among them, B i,0 (ξ) is the B-spline basis function of order 0. When the node value is in the interval [ξ i ,ξ i+1 ], the value of the B-spline basis function is 1, otherwise it is 0, B i,q(ξ) is a B-spline basis function of order q, which is calculated recursively by B-spline basis functions of order q-1. Finally, N ij The calculation method is as follows:

[0056]

[0057] where ω i is the weight of the control point i, and ω j is the weight of the control point j.

[0058] Optionally, step S101 further comprises: constructing a negative thermal expansion metamaterial geometric topology design model according to the following formula:

[0059]

[0060] where ρ (1) is the total density of the first solid material and the second solid material at the control point, ρ (2) is the density of the second solid material at the control point, the proportion of the total density at the control point, M is f(ρ (1) , ρ (2) ) is H is α is the thermal expansion coefficient, is, is k is the stiffness matrix, U is the displacement of the control point, F is the external load applied at the control point, V1 is the volume fraction corresponding to the first solid material, and the upper limit is the lower limit is V2 is the volume fraction corresponding to the second solid material, and the upper limit is the lower limit is K bulk is the equivalent bulk modulus of the microstructure corresponding to the negative thermal expansion metamaterial, K set is the constrained minimum value of the equivalent bulk modulus.

[0061] Specifically, the negative thermal expansion coefficient or zero thermal expansion coefficient of the thermoelastic metamaterial is taken as the optimization objective, the volume fractions V1, V2 of the first solid material and the second solid material respectively, and the minimum bulk modulus K bulk of the metamaterial is taken as the constraint function to construct a negative thermal expansion metamaterial geometric topology design model, and the steps are as follows: an initial microstructure is constructed by using a matlab random function, please refer to Figure 3 which shows a schematic diagram of an initial random density distribution provided by the embodiment of the application, and at the same time, the initial microstructure is ensured to be symmetric about the horizontal, vertical and diagonal symmetry axes to ensure cubic symmetry; the grid dependence and chessboard phenomenon are eliminated by the following formula and by using a density filter and threshold projection of the design variable:

[0062]

[0063] wherein, is the result of the cell density after post-processing by a density filter, is the cell density, is the cell density after threshold projection, r min denotes the radius of a circle centered at cell e and is used to determine the number of cells participating in the cell density filtering, k is used to determine the cell density variable and when k = 1, the cell density variable is determined as when k = 2, the cell density variable is determined as N e is the cell density variable satisfying the following condition the number of cells whose distance to the cell density variable is smaller than a selected minimum radius r min . β k is used to control the strength of the threshold projection and gives a linear interpolation when β k tends to 0 and approaches a step function when β k tends to infinity. η is used to control the position of the inflection point and is set to 0.5, is a weight function and determines the value of k depending on the distance between cell e and cell j, and h is the grid step size and is used to determine the discretization accuracy of the grid.

[0064] In step S102, the relationship interpolation between the multiple sets of density distributions is obtained, and the elastic modulus and the thermal expansion coefficient are obtained according to the relationship interpolation.

[0065] Specifically, the density distribution includes the design variable control point density distribution and the cell density, respectively, and the elastic modulus and the thermal expansion coefficient of different cells are represented based on the relationship interpolation between the design variable control point density distribution and the cell density distribution.

[0066] In step S103, the equivalent elastic matrix corresponding to the thermoelastic metamaterial is calculated according to the elastic modulus, and the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial is calculated according to the thermal expansion coefficient.

[0067] Specifically, the numerical homogenization method based on isogeometric analysis is used to divide the isogeometric grid, and the equivalent elastic matrix corresponding to the thermoelastic metamaterial is calculated according to the elastic modulus, and the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial is calculated according to the thermal expansion coefficient, and the specific steps are as follows: the isogeometric grid division method based on NURBS curve is used, for a two-dimensional example, the number of cells is divided into 100x100, and the periodic boundary condition is applied, please refer to Figure 4A periodic boundary condition application method schematic diagram provided by an embodiment of the application is shown, a unit stiffness matrix is calculated, and an overall stiffness matrix is assembled, as shown in the following formula:

[0068]

[0069] wherein K is an overall stiffness matrix, K e is a unit stiffness matrix, B e is a unit strain-displacement matrix, D e is a unit stress-strain matrix, J1 is a spatial variation relationship from a NURBS parameter domain to a physical domain, J2 is a spatial variation relationship from a NURBS parameter domain to an integral domain, N is a total number of units, V e is an area (in two dimensions) or a volume (in three dimensions) of a unit, and T is a transpose operation of a matrix. Further, a macro global strain field is applied, please refer to Figure 5 A global macro strain field schematic diagram provided by an embodiment of the application is shown. For a two-dimensional case, three direction unit strain fields are applied as follows: ε 1 =[1 0 0] T ,ε 2 =[0 10] T ,ε 3 =[0 0 1] T , and a thermal strain field is: ε 4 =[1 1 0] T In Figure 4 , a first macro global strain field 51 is ε 1 , a second macro global strain field 52 is ε 2 , a third macro global strain field 53 is ε 3 , and a global thermal strain field 54 is ε 4 . A local load generated by the applied unit strain field is calculated, a displacement field corresponding to the unit strain field and a displacement field corresponding to the local load are solved, and the calculation method is as follows.

[0070]

[0071] wherein U 0 is a displacement field corresponding to the unit strain field, U is a displacement field corresponding to the local load, and F is a local load generated by the unit strain field. Then, an equivalent elastic matrix corresponding to the thermoelastic metamaterial is calculated by the following formula:

[0072]

[0073] wherein, is an equivalent elastic matrix, V is a volume fraction corresponding to a solid material, is a global unit strain field, and ε rs ,εrs is the local strain field, D pqrs is the local variable stiffness tensor, which is calculated as follows:

[0074] D pqrs =ED0;

[0075] E is Young's modulus. For the two-dimensional plane stress structure optimization problem, D0 can be obtained as:

[0076]

[0077] Where μ is the Poisson's ratio of the thermoelastic metamaterial. ij and kl represent the two macroscopic dimensions, and pq and rs represent the two microscopic dimensions. The thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial is calculated using the following formula:

[0078]

[0079] in, is the thermal expansion coefficient matrix, α rs is the thermal expansion coefficient at the microscopic scale, and for.

[0080] In step S104, the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix are discretized respectively through isogeometric grids, and a constraint function is constructed according to the equivalent elastic matrix, and an objective function is constructed according to the thermal expansion coefficient matrix. The sensitivity information corresponding to the first design variable is calculated through the objective function and the constraint function; the first design variable is multiple groups of density distributions.

[0081] Specifically, the calculation equations of the equivalent elastic matrix and thermal expansion coefficient matrix are discretized into the following form after isogeometric grid discretization:

[0082]

[0083] A constraint function is constructed based on the equivalent elastic matrix, and an objective function is constructed based on the thermal expansion coefficient matrix. The adjoint method is used to derive and calculate the sensitivity information of the objective function and the constraint function to the first design variable, wherein the design variable is the control point density described by the multi-material interpolation model. The specific steps are as follows: obtain the first partial derivative of the objective function with respect to the first design variable, obtain the second partial derivative of the constraint function with respect to the first design variable; and calculate the sensitivity information corresponding to the first design variable based on the first partial derivative and the second partial derivative. The first partial derivative and the second partial derivative respectively include: the partial derivative of the objective function with respect to the control point density of the design variable, the partial derivative of the thermal expansion coefficient with respect to the design variable, the partial derivative of the equivalent elastic matrix and the equivalent thermal stress matrix with respect to the element density, the partial derivative of the element density with respect to the control point density, the partial derivative of the element single-row modulus and the thermal expansion coefficient with respect to the element density, the partial derivative of the element density with respect to the control point density, and the partial derivative of the equivalent elastic matrix and the equivalent thermal stress matrix with respect to the control point density. The partial derivative of the objective function with respect to the control point density of the design variable can be calculated as follows:

[0084]

[0085] The partial derivative of the thermal expansion coefficient with respect to the design variable can be obtained as follows:

[0086]

[0087] The partial derivatives of the equivalent elastic matrix and the equivalent thermal stress matrix with respect to the element density and the partial derivatives of the element density with respect to the control point density are calculated according to the chain rule. The calculation method is as follows:

[0088]

[0089] According to the multi-material interpolation model, the partial derivatives of the element single-row modulus and thermal expansion coefficient with respect to the element density are as follows:

[0090]

[0091] The partial derivative of the cell density with respect to the control point density is as follows:

[0092]

[0093] The partial derivatives of the equivalent elastic matrix and the equivalent thermal stress matrix with respect to the control point density are as follows:

[0094]

[0095] Among them, S eAfter representing the set of all units affected by the j-th control point, the initial sensitivity information is obtained according to the first and second partial derivatives. Then, the initial sensitivity information is filtered based on density filtering and threshold filtering, so as to obtain the sensitivity information of the unit density after filtering, which is expressed by the following formula:

[0096]

[0097] Step S105 , updating the first design variable according to the sensitivity information until the thermal expansion coefficient meets the convergence condition, and obtaining the second design variable when the convergence condition is met through a geometric topology design model such as a negative thermal expansion metamaterial.

[0098] Specifically, the filtered sensitivity information on the unit density is used as the final sensitivity information, and the sensitivity information and constraint function information are input into the MMA solver to update the first design variable until the final prescribed negative thermal expansion result is obtained and converged.

[0099] Step S106 , optimizing the design of the thermoelastic metamaterial according to the second design variable.

[0100] Specifically, the geometric topology design model of negative thermal expansion metamaterials obtains the design variable at the time of convergence and uses it as the second design variable. The second design variable is the optimal design variable of the thermoelastic metamaterial, and the thermoelastic metamaterial is optimized according to the second design variable. Please refer to Figure 7 , which shows a schematic diagram of an optimized configuration of a thermoelastic metamaterial provided in an embodiment of the present application. The optimized material properties are shown in Table 1 below:

[0101] [Table 1]

[0102]

[0103] The final thermal expansion coefficient of the optimized configuration is -2.050, which meets the negative thermal expansion design requirements. Please refer to Figure 7 , which shows a schematic diagram of a 3×3 periodic arrangement configuration of a negative thermal expansion two-dimensional structure provided in an embodiment of the present application.

[0104] This application adopts the above method to construct a multi-material interpolation model, obtain the density distribution of multiple solid materials in the design domain, and calculate the elastic modulus and thermal expansion coefficient of the material through the interpolation relationship. Then, the equivalent elastic matrix and thermal expansion coefficient matrix of the thermoelastic metamaterial are calculated respectively according to the elastic modulus and thermal expansion coefficient, and the objective function and constraint function are constructed based on this. The design variables are updated in combination with sensitivity analysis until the thermal expansion coefficient meets the convergence conditions, thereby obtaining the optimization results of the negative thermal expansion metamaterial that meets the design requirements. Then, the multi-material interpolation model can accurately represent the complex material distribution and improve the design accuracy. At the same time, combined with the isogeometric topology optimization design, it can not only ensure the consistency of the geometric and physical models, but also flexibly adjust the design under the constraints, thereby achieving efficient and high-precision optimization design of negative thermal expansion or zero thermal expansion metamaterials.

[0105] Please refer to Figure 8 , which shows a module schematic diagram of an isogeometric topology optimization device based on thermoelastic metamaterial design provided by an embodiment of the present application, the device includes a multi-material interpolation model construction module 81 and a thermoelastic metamaterial design module 82, wherein,

[0106] The multi-material interpolation model construction module is used to construct a multi-material interpolation model corresponding to the thermoelastic metamaterial, and obtain multiple groups of density distributions corresponding to multiple solid materials in the thermoelastic metamaterial design domain and one density distribution corresponding to one solid material based on the multi-material interpolation model.

[0107] A thermoelastic metamaterial design module is used to obtain the relationship interpolation between multiple groups of density distributions, and obtain the elastic modulus and thermal expansion coefficient based on the relationship interpolation; calculate the equivalent elastic matrix corresponding to the thermoelastic metamaterial based on the elastic modulus, and calculate the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial based on the thermal expansion coefficient; discretize the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix respectively through equal geometric grids, and construct a constraint function based on the equivalent elastic matrix, and construct an objective function based on the thermal expansion coefficient matrix, and calculate the sensitivity information corresponding to the first design variable through the objective function and the constraint function; the first design variable is multiple groups of density distributions; the first design variable is updated according to the sensitivity information until the thermal expansion coefficient meets the convergence condition, and the second design variable when the convergence condition is met is obtained through a geometric topology design model such as a negative thermal expansion metamaterial; the thermoelastic metamaterial is optimized according to the second design variable.

[0108] In one possible embodiment, the multi-material interpolation model construction module 81 is used to construct a multi-material interpolation model corresponding to the thermoelastic metamaterial, specifically including: dividing the thermoelastic metamaterial into units to obtain a plurality of units; obtaining the Young's modulus corresponding to each unit and obtaining the density distribution corresponding to each unit; and constructing the multi-material interpolation model according to the Young's modulus corresponding to each unit and the density distribution corresponding to each unit using the following formula:

[0109]

[0110] Among them, E ei is the Young's modulus at the i-th unit after interpolation, α ei is the thermal expansion coefficient of the i-th unit after interpolation, is the total density of the first solid material and the second solid material at unit i, and And when When , unit i is completely filled with solid phase, When , the unit i is completely filled with the void phase. is the type of solid at unit i, and when When , only the first solid material exists in unit i, and when When , only the second solid material exists in unit i, N is the number of units, where i ranges from 1 to N, p is the penalty factor, E1 is the Young's modulus corresponding to the first solid material, E2 is the Young's modulus corresponding to the second solid material, α1 is the thermal expansion coefficient corresponding to the first solid material, and α2 is the thermal expansion coefficient corresponding to the second solid material.

[0111] In one possible implementation, the multi-material interpolation model construction module 81 is used to construct a preset basis function and obtain the density at the control point; based on the preset basis function and the density at the control point, the total density and the type of solid are calculated using the following formula:

[0112]

[0113] in, for The density at the corresponding control point, for, for The density at the corresponding control point, is, ic is the geometric center of the i-th unit, c i is the set of control points contained in the i-th unit, c ij c i The j-th control point in N ij is the preset basis function.

[0114] In a possible implementation, the multi-material interpolation model construction module 81 is used to construct a preset basis function, specifically including: constructing a target node vector based on the order of the preset basis function and the number of control points; and constructing the preset basis function according to the target node vector.

[0115] In one possible implementation, the thermoelastic metamaterial design module 82 is configured to calculate the equivalent elastic matrix corresponding to the thermoelastic metamaterial based on the elastic modulus and the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial based on the thermal expansion coefficient. Specifically, the equivalent elastic matrix corresponding to the thermoelastic metamaterial is calculated using the following formula:

[0116]

[0117] in, is the equivalent elastic matrix, V is the volume fraction of the solid material, is the global unit strain field, ε rs ,ε rs is the local strain field, D pqrs is the local variable stiffness tensor, ij and kl represent the two macroscopic dimensions, and pq and rs represent the two microscopic dimensions. The thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial is calculated using the following formula: in, is the thermal expansion coefficient matrix, α rs is the thermal expansion coefficient at the microscopic scale.

[0118] In one possible embodiment, the thermoelastic metamaterial design module 82 is used to calculate the sensitivity information corresponding to the first design variable through the objective function and the constraint function, specifically including: obtaining the first partial derivative of the objective function with respect to the first design variable, obtaining the second partial derivative of the constraint function with respect to the first design variable; and calculating the sensitivity information corresponding to the first design variable based on the first partial derivative and the second partial derivative.

[0119] In a possible implementation, the processing module 22 is configured to construct a geometric topology design model of a negative thermal expansion metamaterial, specifically comprising: constructing a geometric topology design model of a negative thermal expansion metamaterial according to the following formula:

[0120]

[0121] Among them, ρ (1) is the total density of the first solid material and the second solid material at the control point, ρ (2) is the density of the second solid material at the control point, accounting for the proportion of the total density at the control point, M is, f(ρ (1) ,ρ (2) ) is, H is, α is the thermal expansion coefficient, for, where K is the stiffness matrix, U is the displacement of the control point, F is the external load applied at the control point, V1 is the volume fraction corresponding to the first solid material, and its upper bound is The lower bound is V2 is the volume fraction corresponding to the second solid material, and its upper bound is The lower bound is K bulk is the equivalent bulk modulus of the microstructure of the negative thermal expansion metamaterial, K set is the constrained minimum value of the equivalent bulk modulus.

[0122] It should be noted that the above embodiments provide devices that implement their functions using only the division of the above functional modules as examples. In actual applications, the above functions can be assigned to different functional modules as needed, that is, the internal structure of the device can be divided into different functional modules to complete all or part of the functions described above. In addition, the device and method embodiments provided in the above embodiments are based on the same concept. The specific implementation process is detailed in the method embodiment and will not be repeated here.

[0123] This application also provides an electronic device. Figure 9 , Figure 9 9 is a schematic diagram of the structure of an electronic device provided by an embodiment of the present application. The electronic device may include: at least one processor 901, at least one communication bus 902, a user interface 903, at least one network interface 904, and a memory 905.

[0124] The communication bus 902 is used to implement connection and communication between these components.

[0125] The user interface 903 may include a display screen (Display) and a camera (Camera). Optionally, the user interface 903 may also include a standard wired interface and a wireless interface.

[0126] The network interface 904 may optionally include a standard wired interface or a wireless interface (such as a WI-FI interface).

[0127] The processor 901 can include one or more processing cores. The processor 901 connects various parts within the server through various interfaces and lines, and performs various functions of the server and processes data by running or executing instructions, programs, code sets or instruction sets stored in the memory 905, and calling data stored in the memory 905. Alternatively, the processor 901 can be implemented in at least one of a hardware form of a digital signal processing (DSP), a field-programmable gate array (FPGA), and a programmable logic array (PLA). The processor 901 can integrate a combination of one or more of a central processing unit (CPU), a graphics processor (GPU), and a modem. Among them, the CPU is mainly used to process operating systems, user interfaces, and application programs; the GPU is used to render and draw the content to be displayed on the display screen; and the modem is used to process wireless communication. It can be understood that the above-mentioned modem can also not be integrated into the processor 901, but can be realized by a separate chip.

[0128] The memory 905 can include a random access memory (RAM) and can also include a read-only memory (ROM). Optionally, the memory 905 includes a non-transitory computer-readable storage medium. The memory 905 can be used to store instructions, programs, codes, code sets or instruction sets. The memory 905 can include a program storage area and a data storage area, wherein the program storage area can store instructions for implementing an operating system, instructions for at least one function (such as a touch function, a sound playing function, an image playing function, etc.), instructions for implementing the above-mentioned various method embodiments, etc.; the data storage area can store data involved in the above-mentioned various method embodiments, etc. The memory 905 can also be at least one storage device located away from the aforementioned processor 901. For reference Figure 9 The memory 905 as a computer storage medium can include an operating system, a network communication module, a user interface module, and an application program.

[0129] In Figure 9In the electronic device shown, the user interface 903 is mainly used to provide an input interface for the user and obtain the data input by the user; and the processor 901 can be used to call the application stored in the memory 905. When executed by one or more processors 901, the electronic device executes one or more of the methods described in the above embodiments. It should be noted that for the aforementioned method embodiments, for the sake of simplicity of description, they are all expressed as a series of action combinations, but those skilled in the art should know that this application is not limited to the described order of actions, because according to this application, certain steps can be performed in other orders or simultaneously. Secondly, those skilled in the art should also know that the embodiments described in the specification are all preferred embodiments, and the actions and modules involved are not necessarily required for this application.

[0130] The present application also provides a computer-readable storage medium storing instructions, which, when executed by one or more processors, enable an electronic device to execute one or more of the methods described in the above embodiments.

[0131] In the above embodiments, the description of each embodiment has its own focus. For parts that are not described in detail in a certain embodiment, reference can be made to the relevant descriptions of other embodiments.

[0132] In the several embodiments provided in this application, it should be understood that the disclosed devices can be implemented in other ways. For example, the device embodiments described above are merely schematic, such as the division of units, which is only a logical function division. In actual implementation, there may be other division methods, such as multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some service interface, and the indirect coupling or communication connection of devices or units can be electrical or other forms.

[0133] Units described as separate components may or may not be physically separate, and components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of these units may be selected to achieve the purpose of this embodiment according to actual needs.

[0134] In addition, the functional units in the various embodiments of the present application may be integrated into a single processing unit, or each unit may exist physically separately, or two or more units may be integrated into a single unit. The aforementioned integrated units may be implemented in the form of hardware or software functional units.

[0135] The integrated unit, if implemented in the form of a software function unit and sold or used as an independent product, can be stored in a computer readable memory. Based on such understanding, the technical solutions of the present application essentially or say the part that contributes to the prior art or the whole or part of the technical solutions can be embodied in the form of a software product. The computer software product is stored in a memory and includes a plurality of instructions for causing a computer device (which can be a personal computer, a server or a network device, etc.) to execute all or part of the steps of the embodiments of the present application. The aforementioned memory includes: a U disk, a mobile hard disk, a magnetic disk or an optical disk and various program code storage media.

[0136] The above is only exemplary embodiments of the present application, and cannot limit the scope of the present application. That is, any equivalent changes and modifications made in accordance with the teachings of the present application are still within the scope of the present application. Other embodiments of the present application will be readily apparent to those skilled in the art upon considering the specification and the practical true disclosure.

[0137] The present application is intended to cover any variations, uses or adaptive changes of the present application, which follow the general principles of the present application and include common knowledge or conventional technical means in the art not disclosed in the present application.

Claims

1. An isogeometric topology optimization method based on thermoelastic metamaterial design, characterized in that: The method comprises: Constructing a multi-material interpolation model corresponding to the thermoelastic metamaterial, and obtaining multiple groups of density distributions corresponding to multiple solid materials in the design domain of the thermoelastic metamaterial according to the multi-material interpolation model, one density distribution corresponding to each solid material; Obtaining a relationship interpolation value between a plurality of groups of density distributions, and obtaining an elastic modulus and a thermal expansion coefficient according to the relationship interpolation value; Calculating an equivalent elastic matrix corresponding to the thermoelastic metamaterial according to the elastic modulus, and calculating a thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial according to the thermal expansion coefficient; The calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix are respectively discretized through an isogeometric grid, and a constraint function is constructed according to the equivalent elastic matrix, and an objective function is constructed according to the thermal expansion coefficient matrix, and sensitivity information corresponding to a first design variable is calculated through the objective function and the constraint function, specifically including: obtaining a first partial derivative of the objective function with respect to the first design variable, obtaining a second partial derivative of the constraint function with respect to the first design variable; calculating sensitivity information corresponding to the first design variable according to the first partial derivative and the second partial derivative; the first design variable is a plurality of groups of density distributions; updating the first design variable according to the sensitivity information until the thermal expansion coefficient satisfies a convergence condition, and obtaining a second design variable when the convergence condition is satisfied through a geometric topology design model such as a negative thermal expansion metamaterial; Optimizing the design of the thermoelastic metamaterial according to the second design variable; The method of constructing a multi-material interpolation model corresponding to the thermoelastic metamaterial specifically includes: dividing the thermoelastic metamaterial into units to obtain a plurality of units; obtaining the Young's modulus corresponding to each unit and obtaining the density distribution corresponding to each unit; and constructing the multi-material interpolation model according to the Young's modulus corresponding to each unit and the density distribution corresponding to each unit using the following formula: ； in, After interpolation Young's modulus at each element, After interpolation The thermal expansion coefficient at each unit, For unit The total density of the first solid material and the second solid material, and , and when When, unit is completely filled with solid phase. When the unit is completely filled with the void phase, For unit The type of solid, and when When, unit When only the first solid material is present When, unit only the second solid material is present, is the number of units, where The value range is 1 to , is the penalty factor, is the Young's modulus corresponding to the first solid material, is the Young's modulus corresponding to the second solid material, is the thermal expansion coefficient of the first solid material, is the thermal expansion coefficient of the second solid material.

2. The method according to claim 1, characterized in that The method further comprises: Construct the preset basis function and obtain the density at the control point; According to the preset basis function and the density at the control point, the total density and the type of the solid are calculated using the following formula: ； in, for The density at the corresponding control point, for The density at the corresponding control point, For the The geometric center of the unit, For the The set of control points contained in each cell, is the preset basis function.

3. The method according to claim 2, characterized in that The construction of the preset basis function specifically includes: Constructing a target node vector based on the order of the preset basis function and the number of control points; The preset basis function is constructed according to the target node vector.

4. The method according to claim 1, wherein Calculating the equivalent elastic matrix corresponding to the thermoelastic metamaterial according to the elastic modulus, and calculating the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial according to the thermal expansion coefficient, specifically includes: The equivalent elastic matrix corresponding to the thermoelastic metamaterial is calculated by the following formula: ； in, is the equivalent elastic matrix, is the volume fraction corresponding to the solid material, 、 is the global unit strain field, is the local strain field, is the local variable stiffness tensor, and Representing the two macroscopic dimensions, and They represent two dimensions of microscopic view respectively; The thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial is calculated by the following formula: ； in, is the thermal expansion coefficient matrix, is the thermal expansion coefficient at the microscopic scale.

5. The method according to claim 2, characterized in that Constructing a geometric topology design model of the negative thermal expansion metamaterial, etc., specifically includes: The geometric topology design model of the negative thermal expansion metamaterial is constructed according to the following formula: ； in, is the total density of the first solid material and the second solid material at the control point, is the ratio of the density of the second solid material at the control point to the total density at the control point, is the thermal expansion coefficient, is the stiffness matrix, is the displacement of the control point, is the external load applied at the control point, is the volume fraction corresponding to the first solid material, and its upper bound is , the lower bound is , is the volume fraction corresponding to the second solid material, and its upper bound is , the lower bound is , is the equivalent bulk modulus of the microstructure corresponding to the negative thermal expansion metamaterial, is the constrained minimum value of the equivalent bulk modulus.

6. An isogeometric topology optimization device based on thermoelastic metamaterial design, characterized in that: The device includes a multi-material interpolation model building module and a thermoelastic metamaterial design module, wherein, The multi-material interpolation model construction module is used to construct a multi-material interpolation model corresponding to the thermoelastic metamaterial, and obtain multiple groups of density distributions corresponding to multiple solid materials in the thermoelastic metamaterial design domain according to the multi-material interpolation model, one density distribution corresponding to each solid material; The thermoelastic metamaterial design module is used to obtain the relationship interpolation between multiple groups of density distributions, and obtain the elastic modulus and thermal expansion coefficient based on the relationship interpolation; calculate the equivalent elastic matrix corresponding to the thermoelastic metamaterial based on the elastic modulus, and calculate the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial based on the thermal expansion coefficient; discretize the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix respectively through equigeometric grids, and construct a constraint function based on the equivalent elastic matrix, and construct an objective function based on the thermal expansion coefficient matrix, and calculate the sensitivity information corresponding to the first design variable through the objective function and the constraint function, specifically including: obtaining the first partial derivative of the objective function with respect to the first design variable, obtaining the second partial derivative of the constraint function with respect to the first design variable; according to the first Partial derivative and the second partial derivative, calculate the sensitivity information corresponding to the first design variable; the first design variable is a plurality of groups of density distributions; the first design variable is updated according to the sensitivity information until the thermal expansion coefficient meets the convergence condition, and the second design variable when the convergence condition is met is obtained through a geometric topology design model such as a negative thermal expansion metamaterial; the thermoelastic metamaterial is optimized according to the second design variable; the multi-material interpolation model corresponding to the thermoelastic metamaterial is constructed, specifically including: dividing the thermoelastic metamaterial into units to obtain a plurality of units; obtaining the Young's modulus corresponding to each unit, and obtaining the density distribution corresponding to each unit; according to the Young's modulus corresponding to each unit and the density distribution corresponding to each unit, the multi-material interpolation model is constructed by the following formula: ； in, After interpolation Young's modulus at each element, After interpolation The thermal expansion coefficient at each unit, For unit The total density of the first solid material and the second solid material, and , and when When, unit is completely filled with solid phase. When the unit is completely filled with the void phase, For unit The type of solid, and when When, unit When only the first solid material is present When, unit only the second solid material is present, is the number of units, where The value range is 1 to , is the penalty factor, is the Young's modulus corresponding to the first solid material, is the Young's modulus corresponding to the second solid material, is the thermal expansion coefficient of the first solid material, is the thermal expansion coefficient of the second solid material.

7. An electronic device, characterized in that: The electronic device comprises a processor, a communication bus, a user interface, a network interface and a memory, wherein the memory is used to store instructions, the user interface and the network interface are used to communicate with other devices, and the processor is used to execute the instructions stored in the memory so that the electronic device executes the method according to any one of claims 1 to 5.

8. A computer-readable storage medium, characterized in that The computer-readable storage medium stores instructions, and when the instructions are executed, the method according to any one of claims 1 to 5 is performed.

Images