Isogeometric topology optimization method and device based on thermoelastic metamaterial design

Abstract

The invention provides an isogeometric topology optimization method and device based on thermoelastic metamaterial design, and relates to the field of isogeometric topology optimization. The method comprises the following steps: constructing a multi-material interpolation model corresponding to a thermoelastic metamaterial, and obtaining multiple groups of density distributions corresponding to multiple solid materials and one density distribution corresponding to one solid material in a thermoelastic metamaterial design domain according to the multi-material interpolation model; obtaining a relation interpolation among the multiple groups of density distributions, and obtaining an elastic modulus and a thermal expansion coefficient according to the relation interpolation; 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; performing discretization form processing on the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix through an isogeometric grid, constructing a constraint function according to the equivalent elastic matrix, constructing a target function according to the thermal expansion coefficient matrix, and calculating sensitivity information corresponding to the first design variable through the target function and the constraint function; updating the first design variable according to the sensitivity information until the thermal expansion coefficient meets a convergence condition, and acquiring a second design variable meeting the convergence condition through geometric topology design models such as a negative thermal expansion metamaterial; and performing optimization design on the thermoelastic metamaterial according to the second design variable. The problem that when a complex geometric structure is processed, a finite element method is difficult to guarantee efficient and high-precision design of the thermoelastic metamaterial is solved.

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] At present, 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 metamaterials. Topological optimization uses optimization algorithms to find the optimal topological layout within the unit cell, which can greatly improve the design freedom and customize the design of metamaterials with different shapes while meeting certain constraints. Therefore, it is necessary to consider the topological optimization design scheme of thermoelastic metamaterials.

[0003] Traditional topology optimization design methods mainly use the finite element method (FEM) for numerical analysis. In this method, the finite element mesh is only an approximation of the design domain geometry rather than an accurate description, and the response between adjacent finite elements only satisfies C0 continuity, that is, the displacement field may be discontinuous or non-smooth at the unit boundary. Since FEM meshes are usually polygons or polyhedrons, this geometric approximation will lead to a decrease in calculation accuracy, especially when dealing with complex geometric structures. The finite element method is difficult to ensure efficient and high-precision design of thermoelastic metamaterials.

[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 an isogeometric topology optimization method and device 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, an isogeometric topology optimization method based on thermoelastic metamaterial design is provided, the method comprising: 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 and one density distribution corresponding to one solid material according to the multi-material interpolation model; obtaining the relationship interpolation between the multiple groups of density distributions, and obtaining the elastic modulus and thermal expansion coefficient according to the relationship interpolation; 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; discretizing the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix respectively through isogeometric grids, and constructing a constraint function according to the equivalent elastic matrix, and 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; 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 meets the convergence condition, and obtaining the second design variable when the convergence condition is met through the isogeometric topology design model of the negative thermal expansion metamaterial; optimizing the design of the thermoelastic metamaterial according to 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 by 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 , 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, E 1 is the Young's modulus corresponding to the first solid material, E 2 is the Young's modulus corresponding to the second solid material, α 1 is the thermal expansion coefficient of the first solid material, α2 is the corresponding thermal expansion coefficient of 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 by 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 ith unit, c ij c i The jth 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 by 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, pq and rs represent the two microscopic dimensions, and the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial is calculated by the following formula: in, is the thermal expansion coefficient matrix, α rs is the coefficient of thermal expansion 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 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; 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 topological design model of a negative thermal expansion metamaterial specifically includes: constructing a geometric topological 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, is, K is the stiffness matrix, U is the displacement of the control point, F is the external load applied at the control point, V 1 is the volume fraction corresponding to the first solid material, and its upper bound is The lower bound is V 2 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 building 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 according to the multi-material interpolation model.

[0023] 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; update the first design variable according to the sensitivity information until the thermal expansion coefficient meets the convergence condition, and obtain the second design variable when the convergence condition is met through a geometric topology design model such as a negative thermal expansion metamaterial; optimize the design of the thermoelastic metamaterial 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 executes any one 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 the present 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 the interpolation relationship. Then, the equivalent elastic matrix and thermal expansion coefficient matrix of the thermoelastic metamaterial are calculated according to the elastic modulus and thermal expansion coefficient, 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 distribution of complex materials 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 to achieve efficient and high-precision optimization design of negative thermal expansion or zero thermal expansion metamaterials.

[0028] 2. Divide 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 nature of the thermoelastic metamaterial can be better described, and the accuracy and flexibility of the thermoelastic metamaterial design can be improved.

[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, thereby achieving the effect of accurately controlling the material distribution, optimizing the material's negative thermal expansion characteristics, and maximizing the material performance under given constraints. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 It 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 It 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 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 It is a schematic diagram of a global macroscopic strain field provided in an embodiment of the present application;

[0035] Figure 6 It 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 It is a module schematic diagram of an isogeometric topology optimization device based on thermoelastic metamaterial design provided in an embodiment of the present application;

[0038] Fig. 9 It is a structural schematic 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 technicians in this field 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 embodiments of the present application are only for the purpose of describing specific embodiments, and are not intended to be used as limitations to the present application. As used in the specification of the present application, the singular expressions "one", "a kind of", "said", "above", "the" and "this" are intended to also include plural expressions, unless there is a clear indication to the contrary in the context. 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 listed items.

[0042] In the following, the terms "first" and "second" are used for descriptive purposes only and are not to be understood as suggesting or implying relative importance or implicitly indicating the number of the indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of the features, and in the description of the embodiments of the present 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] At present, 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 metamaterials. Topological optimization uses optimization algorithms to find the optimal topological layout within the unit cell, which can greatly improve the design freedom and customize the design of metamaterials with different shapes while meeting certain constraints. Therefore, it is necessary to consider the topological optimization design scheme of thermoelastic metamaterials.

[0045] Traditional topology optimization design methods mainly use the finite element method (FEM) for numerical analysis. In this method, the finite element mesh is only an approximation of the design domain geometry rather than an accurate description, and the response between adjacent finite elements only satisfies C0 continuity, that is, the displacement field may be discontinuous or non-smooth at the unit boundary. Since FEM meshes are usually polygons or polyhedrons, this geometric approximation will lead to a decrease in calculation accuracy, especially when dealing with complex geometric structures. The finite element method is difficult to ensure efficient and high-precision design of thermoelastic metamaterials.

[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 schematic flow chart of an isogeometric topology optimization method based on thermoelastic metamaterial design provided in an embodiment of the present application, and the flow chart mainly includes the following steps: S101 to S106.

[0048] Step S101 : 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 design domain of the thermoelastic metamaterial 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 in the thermoelastic microstructure design domain, and multiple groups of density distributions corresponding to multiple solid materials in the thermoelastic metamaterial design domain are obtained according to the multi-material interpolation model, and one density distribution corresponding to one solid material. The specific construction method of the multi-material interpolation model is as follows: the thermoelastic metamaterial is divided 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; 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:

[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, E 1 is the Young's modulus corresponding to the first solid material, E 2 is the Young's modulus corresponding to the second solid material, and E 1 =1,E 2 =1,α 1 is the thermal expansion coefficient of the first solid material, α 2 is the thermal expansion coefficient of the second solid material. and The calculation is performed in the following way: 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:

[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 jth 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 in 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={ξ 1 ,ξ 2 ,...,ξ m+q+1}, Ξ is the target node vector, {ξ 1 ,ξ 2 ,...,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, Bi,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. i,q (ξ) is the B-spline basis function of order q, which is obtained by recursive calculation of the B-spline basis function of order q-1. Finally, N is obtained. ij The calculation method is:

[0056]

[0057] Among them, ω i is the weight of control point i, ω j is the weight of control point j.

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

[0059]

[0060] 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, is, k is the stiffness matrix, U is the displacement of the control point, F is the external load applied at the control point, V 1 is the volume fraction corresponding to the first solid material, and its upper bound is The lower bound is V 2 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.

[0061] Specifically, taking the negative thermal expansion coefficient or zero thermal expansion coefficient of the thermoelastic metamaterial as the optimization target, the volume fractions V of the first solid material and the second solid material are respectively 1 ,V 2 and the minimum bulk modulus K of the metamaterial bulk To construct a geometric topology design model for negative thermal expansion metamaterials and other constraint functions, the steps are as follows: Use MATLAB random function to construct the initial microstructure, please refer to Figure 3, which shows a schematic diagram of an initial random density distribution provided by an embodiment of the present application, while ensuring that the initial microstructure is symmetrical about the horizontal, vertical, and diagonal symmetry axes to ensure cubic symmetry; the following formula is used, and the density filter and threshold projection of the design variables are used to eliminate grid dependence and checkerboard phenomenon:

[0062]

[0063] in, is the result of post-processing the cell density through the density filter, for The cell density after threshold projection, r min Represents the radius of a circle centered on unit e and is used to determine the number of units involved in unit density filtering. k is used to determine the unit density variable And when k = 1, determine for When k = 2, determine for N e is a cell density variable that satisfies the following conditions Number of: Cell density variables With cell density variables The distance is less than the selected minimum radius r min . β k Used to control the intensity of the threshold projection, and when β k When it approaches 0, linear interpolation is given; when β k When it approaches infinity, it approaches a step function. η is used to control the position of the inflection point and is set to 0.5. is a weight function, and the value of k is determined according to the distance between unit e and unit j, and h is the grid step size, which is used to determine the discrete accuracy of the grid.

[0064] Step S102, obtaining a relationship interpolation between multiple groups of density distributions, and obtaining an elastic modulus and a thermal expansion coefficient according to the relationship interpolation.

[0065] Specifically, the density distribution includes controlling point density distribution and unit density based on design variables, and interpolating elastic modulus and thermal expansion coefficient of different units based on the relationship between the control point density distribution and the unit density distribution.

[0066] Step S103, 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.

[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. The specific steps are as follows: The isogeometric grid division method based on NURBS curve is used. For the two-dimensional example, the number of units is divided into 100×100, and periodic boundary conditions are applied. Please refer to Figure 4 , which shows a schematic diagram of a method for applying periodic boundary conditions provided in an embodiment of the present application, calculates the unit stiffness matrix, and assembles the overall stiffness matrix, as shown in the following formula:

[0068]

[0069] Among them, K is the overall stiffness matrix, K e is the unit stiffness matrix, B e is the unit strain-displacement matrix, D e is the unit stress-strain matrix, J 1 is the spatial variation relationship from the NURBS parameter domain to the physical domain, J 2 is the spatial variation relationship from the NURBS parameter domain to the integral domain, N is the total number of units, V e is the area (in two dimensions) or volume (in three dimensions) of the unit cell, and T is the transpose operation of the matrix. Further, to apply a macroscopic global strain field, please refer to Figure 5 , which shows a schematic diagram of a global macroscopic strain field provided in an embodiment of the present application. For a two-dimensional case, the unit strain field applied in three directions is: ε 1 =[1 0 0] T ,ε 2 =[0 10] T ,ε 3 =[0 0 1] T , the thermal strain field is: ε 4 =[1 1 0] T ,exist Figure 4 In the first macroscopic global strain field 51, 1 , the second macroscopic global strain field 52 is ε 2 , the third macroscopic global strain field 53 is ε 3 , the global thermal strain field 54 is ε 4 . Calculate the local load generated by applying the unit strain field, and solve the displacement field corresponding to the unit strain field and the displacement field corresponding to the local load. The calculation method is as follows.

[0070]

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

[0072]

[0073] 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, calculated as follows:

[0074] D pqrs =ED 0 ;

[0075] E is Young's modulus. For the two-dimensional plane stress structure optimization problem, D can be obtained. 0 for:

[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 by 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] Step S104, discretizing the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix respectively through isogeometric grids, 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; the first design variable is a plurality of 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 mesh discretization:

[0082]

[0083] A constraint function is constructed according to the equivalent elastic matrix, an objective function is constructed according to the thermal expansion coefficient matrix, and 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, and the specific steps are as follows: obtain the first partial derivative of the objective function to the first design variable, obtain the second partial derivative of the constraint function to the first design variable; calculate the sensitivity information corresponding to the first design variable according to the first partial derivative and the second partial derivative. Wherein, the first partial derivative and the second partial derivative respectively include: the partial derivative of the objective function to the control point density of the design variable, the partial derivative of the thermal expansion coefficient to the design variable, the partial derivative of the equivalent elastic matrix and the equivalent thermal stress matrix to the unit density, the partial derivative of the unit density to the control point density, the partial derivative of the unit single-row modulus and the thermal expansion coefficient to the unit density, the partial derivative of the unit density to the control point density, the partial derivative of the equivalent elastic matrix and the equivalent thermal stress matrix to the control point density, wherein the partial derivative of the objective function 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 unit single-row modulus and thermal expansion coefficient with respect to the unit density are obtained 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 jth control point, the initial sensitivity information is obtained according to the first partial derivative and the second partial derivative. 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, and it 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, thereby updating 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 topological design model such as the negative thermal expansion metamaterial 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] The present 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, so as to achieve the optimization design of efficient and high-precision 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 in 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 according to the multi-material interpolation model.

[0107] 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; update the first design variable according to the sensitivity information until the thermal expansion coefficient meets the convergence condition, and obtain the second design variable when the convergence condition is met through a geometric topology design model such as a negative thermal expansion metamaterial; optimize the design of the thermoelastic metamaterial according to the second design variable.

[0108] In a possible implementation, the multi-material interpolation model construction module 81 is used to construct a multi-material interpolation model corresponding to the thermoelastic metamaterial, specifically including: performing unit segmentation on the thermoelastic metamaterial 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 by 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 , 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, E 1 is the Young's modulus corresponding to the first solid material, E 2 is the Young's modulus corresponding to the second solid material, α 1 is the thermal expansion coefficient of the first solid material, α 2 is the corresponding thermal expansion coefficient of the second solid material.

[0111] In a 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; according to the preset basis function and the density at the control point, the total density and the type of solid are calculated by 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 jth 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 a possible implementation, the thermoelastic metamaterial design module 82 is used to calculate the equivalent elastic matrix corresponding to the thermoelastic metamaterial according to the elastic modulus, and calculate the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial according to the thermal expansion coefficient, specifically including: calculating the equivalent elastic matrix corresponding to the thermoelastic metamaterial by 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, pq and rs represent the two microscopic dimensions, and the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial is calculated by the following formula: in, is the thermal expansion coefficient matrix, α rs is the coefficient of thermal expansion at the microscopic scale.

[0118] In a possible implementation, 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 used to construct a geometric topology design model of a negative thermal expansion metamaterial, specifically including: 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, is, K is the stiffness matrix, U is the displacement of the control point, F is the external load applied at the control point, V 1 is the volume fraction corresponding to the first solid material, and its upper bound is The lower bound is V 2 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: when the device provided in the above embodiment realizes its function, only the division of the above functional modules is used as an example. In actual application, the above functions can be assigned to different functional modules as needed, that is, the internal structure of the device is 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 embodiment belong to the same concept, and the specific implementation process is detailed in the method embodiment, which will not be repeated here.

[0123] The present application also provides an electronic device. Fig. 9 , Fig. 9 9 is a schematic diagram of the structure of an electronic device provided in 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 realize the connection and communication between these components.

[0125] The user interface 903 may include a display screen (Display) and a camera (Camera), and the optional 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] Among them, the processor 901 may include one or more processing cores. The processor 901 uses various interfaces and lines to connect various parts in the entire server, and executes 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. Optionally, the processor 901 can be implemented in at least one hardware form of digital signal processing (Digital Signal Processing, DSP), field programmable gate array (Field-Programmable Gate Array, FPGA), and programmable logic array (Programmable Logic Array, PLA). The processor 901 can integrate one or more combinations of a central processing unit (Central Processing Unit, CPU), a graphics processing unit (Graphics Processing Unit, GPU) and a modem. Among them, the CPU mainly processes the operating system, user interface and application programs; the GPU is responsible for rendering and drawing the content to be displayed on the display screen; the modem is used to process wireless communications. It can be understood that the above-mentioned modem may not be integrated into the processor 901, and it can be implemented by a single chip.

[0128] Among them, the memory 905 may include a random access memory (Random Access Memory, RAM) and may also include a read-only memory (Read-Only Memory). 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 may include a program storage area and a data storage area, wherein the program storage area may store instructions for implementing an operating system, instructions for at least one function (such as a touch function, a sound playback function, an image playback function, etc.), instructions for implementing the above-mentioned various method embodiments, etc.; the data storage area may store data involved in the above-mentioned various method embodiments, etc. The memory 905 may also be optionally at least one storage device located away from the aforementioned processor 901. Refer to Fig. 9 , the memory 905 as a computer storage medium may include an operating system, a network communication module, a user interface module and a xxxxxxxx application program.

[0129] exist Fig. 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 xxxxxxxx application stored in the memory 905, and 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 simple description, they are all expressed as a series of action combinations, but those skilled in the art should know that the present application is not limited by the described order of actions, because according to the present 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 the present application.

[0130] The present application also provides a computer-readable storage medium, which stores instructions. When executed by one or more processors, the electronic device executes one or more of the methods described in the above embodiments.

[0131] In the above embodiments, the description of each embodiment has its own emphasis. 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 only schematic, such as the division of units, which is only a logical function division. There may be other division methods in actual implementation, 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 interfaces, and the indirect coupling or communication connection of devices or units can be electrical or other forms.

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

[0134] In addition, each functional unit in each embodiment of the present application may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit. The above-mentioned integrated unit may be implemented in the form of hardware or in the form of software functional units.

[0135] If the integrated unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable memory. Based on this understanding, the technical solution of the present application, or the part that contributes to the prior art, or all or part of the technical solution can be embodied in the form of a software product, which is stored in a memory and includes several instructions for a computer device (which can be a personal computer, server or network device, etc.) to perform all or part of the steps of the various embodiments of the present application. The aforementioned memory includes: various media that can store program codes, such as USB flash drives, mobile hard drives, magnetic disks or optical disks.

[0136] The above are only exemplary embodiments disclosed in this application, and the scope of the disclosure of this application cannot be limited by this. That is, any equivalent changes and modifications made according to the teachings disclosed in this application are still within the scope of the disclosure of this application. After considering the disclosure of the specification and the truth of practice, those skilled in the art will easily think of other implementation schemes disclosed in this application.

[0137] This application is intended to cover any modifications, uses or adaptive changes disclosed in this application, which follow the general principles disclosed in this application and include common knowledge or customary technical means in the technical field not recorded in this 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 one solid material; Obtaining a relationship interpolation between a plurality of groups of density distributions, and obtaining an elastic modulus and a thermal expansion coefficient according to the relationship interpolation; 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 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, and sensitivity information corresponding to a first design variable is calculated through the objective function and the constraint function; the first design variable is a plurality of groups of density distributions; Performing an updating operation on the first design variable according to 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 a geometric topological design model such as a negative thermal expansion metamaterial; The thermoelastic metamaterial is optimized and designed according to the second design variable.

2. The method according to claim 1, characterized in that The constructing of a multi-material interpolation model corresponding to the thermoelastic metamaterial specifically includes: Performing unit division on the thermoelastic metamaterial to obtain a plurality of units; Obtain the Young's modulus corresponding to each unit, and obtain 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: Among them, E ei is the Young's modulus at the i-th unit after interpolation, α ei is the thermal expansion coefficient at 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 , unit i only contains the first solid material, when , unit i only contains the second solid material, N is the number of units, wherein 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.

3. The method according to claim 2, 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 by the following formula: 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 ith unit, c ij c i The jth control point in N ij is the preset basis function.

4. The method according to claim 3, characterized in that The constructing 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.

5. The method according to claim 1, characterized in that: The step of 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, V is the volume fraction corresponding to 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, pq and rs represent the two microscopic dimensions; The thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial is calculated by the following formula: in, is the thermal expansion coefficient matrix, α rs is the thermal expansion coefficient at the microscopic scale.

6. The method according to claim 3, characterized in that: The calculating the sensitivity information corresponding to the first design variable by using the objective function and the constraint function specifically includes: Obtaining a first partial derivative of the objective function with respect to the first design variable, and obtaining a second partial derivative of the constraint function with respect to the first design variable; Sensitivity information corresponding to a first design variable is calculated according to the first partial derivative and the second partial derivative.

7. The method according to claim 3, characterized in that Constructing a geometric topological design model of the negative thermal expansion metamaterial, etc., specifically includes: The geometric topological design model of the negative thermal expansion metamaterial is constructed according to the following formula: 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 the, 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 V1 U , the lower bound is V1 L , V2 is the volume fraction corresponding to the second solid material, and its upper limit is V2 U , the lower bound 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.

8. An isogeometric topology optimization device based on thermoelastic metamaterial design, characterized in that: The device comprises 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 design domain of the thermoelastic metamaterial according to the multi-material interpolation model, one density distribution corresponding to one 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 according to the relationship interpolation; calculate the equivalent elastic matrix corresponding to the thermoelastic metamaterial according to the elastic modulus, and calculate the thermal expansion coefficient matrix corresponding to the thermoelastic metamaterial according to the thermal expansion coefficient; discretize the calculation equations of the equivalent elastic matrix and the thermal expansion coefficient matrix respectively through isogeometric grids, and construct a constraint function according to the equivalent elastic matrix, and construct an objective function according to 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; update the first design variable according to the sensitivity information until the thermal expansion coefficient meets the convergence condition, and obtain the second design variable when the convergence condition is met through the geometric topology design model such as the negative thermal expansion metamaterial; optimize the design of the thermoelastic metamaterial according to the second design variable.

9. An electronic device, characterized in that: It includes 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 7.

10. 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 7 is performed.

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