Equi-biased anisotropic periodic structure material multi-scale topology optimization method

Abstract

The application discloses an equal geometry-based multi-scale topological optimization method for anisotropic periodic structure material, and comprises the following steps: (1) modeling according to an equal geometry analysis method to obtain discrete point information of a macro periodic structure and various microstructure design domains, and dividing a design subdomain for the macro periodic structure design domain; (2) constructing a coordinate conversion relationship between a natural coordinate system and a material coordinate system in an anisotropic material; (3) performing periodic material distribution optimization to determine the position of various microstructures in the macro periodic structure and the volume fraction of various microstructures by normalizing the relative density; and (4) performing periodic structure / material multi-scale parallel topological optimization to determine the macro periodic structure, various microstructures and equivalent elastic matrices thereof. The application performs multi-scale topological optimization for an anisotropic periodic structure / material based on equal geometry analysis, and is accurate, stable and efficient in convergence, and can improve the structure performance from the macro and micro scales and the structure and material angles.

Description

Technical Field

[0001] This invention belongs to the field of optimization design in computer-aided engineering, specifically relating to a multi-scale topology optimization method for anisotropic periodic structural materials based on isogeometry. Background Technology

[0002] Periodic porous composite structures, due to their unique configurations and multifunctional physical properties such as sound absorption, heat insulation, vibration reduction, impact protection, and lightweight yet high strength, have been widely used in machinery, automotive, and aerospace fields. However, most periodic composite structures designed based on experience are generally not optimal, and with the application and promotion of various high-performance heterogeneous anisotropic composite materials, the design difficulty of periodic composite structures with excellent performance is increasing daily. Furthermore, with the continuous expansion of industrial product applications, the demands for structural performance and reliability from various complex working environments (such as high temperature, high pressure, high speed, and vacuum) are also increasing. This makes it increasingly difficult for purely macroscopic periodic composite structures to meet the needs of high-tech industries for "lightweight structure," "functional specialization," and "performance integration" in industrial products. Therefore, adopting appropriate and efficient methods to introduce microscopic material microstructure design on the basis of macroscopic periodic structure design will be an effective means to further improve the performance of periodic composite structures.

[0003] Topology optimization, as an emerging branch of structural optimization, is a computational design method that automatically generates material layouts that maximize performance under relevant design specifications. It can solve complex structural design problems that traditional trial-and-error methods cannot effectively address, providing a powerful system design strategy for multi-scale periodic structures of macro- and micro-composite materials. Simultaneously, the rapidly developing additive manufacturing technology provides a convenient way to manufacture complex topological structures. Currently, mainstream topology optimization methods include: Homogenization Method, Variable Density Method, Level Set Method (LSM), Phase Field Method, Evolutionary Structural Optimization (ESO), and Moving Morphable Components (MMC). Among these, the Variable Density Method, represented by the Penalized Solid Isotropic Microstructures with Penalization (SIMP), is the most widely used due to its simple model and clear topology. Commonly used performance analysis methods include: Finite Element Method (FEM), Boundary Element Method (BEM), Finite Difference Method (FDM), Finite Volume Method (FVM), Element-free Method, and Isogeometric Analysis (IGA). The first four methods inevitably simplify the model during the conversion from the geometric model to the analytical model. This inconsistency between the geometric and analytical models can lead to accuracy loss in subsequent calculations, and converting complex geometric models into analytical models is also quite challenging. Isogeometric analysis, by using spline functions describing the geometric shape as shape functions of the solution domain, unifies the geometric and analytical models. Furthermore, the high-order continuous spline functions and accurate geometric description ensure the accuracy of the analytical calculations. Therefore, the topology optimization method combining SIMP and isogeometric analysis is an effective approach for designing anisotropic periodic structures / multi-scale material structures.

[0004] The topology optimization design of microstructures with specific properties originated from the inverse homogenization design model proposed by Sigmund in the 1990s. In the following two decades, topology optimization methods have been used to design numerous metamaterial microstructures with specific properties such as maximum bulk modulus, maximum shear modulus, negative Poisson's ratio, and negative thermal expansion coefficient. During this period, researchers also combined microstructure topology optimization design with macrostructure topology optimization design, proposing multi-scale topology optimization design models that consider single and multiple types of microstructures, making materials design more practically valuable. However, most existing research on microstructure design and macro / micro multi-scale topology optimization design schemes are based on homogeneous isotropic materials, focusing solely on improving structural performance by deeply exploring the potential of material distribution from a "structural" perspective. Research on microstructure design and multi-scale material / structure design of high-performance anisotropic composite materials with good designability is relatively scarce. Furthermore, no research results introducing macro-periodic structural design into the multi-scale structural / material design framework have been publicly reported.

[0005] Against the above background, this invention proposes a multi-scale topology optimization method for anisotropic periodic structures / materials based on isogeometric analysis. This method can improve the structural performance of products from both macroscopic and microscopic scales and from both structural and material perspectives while ensuring the macroscopic periodicity of the structure. Summary of the Invention

[0006] Given the non-optimal performance of most current empirically designed periodic porous composite structures and the lack of research on multi-scale topology optimization design of periodic structures / materials for anisotropic composites, the technical problem to be solved by this invention is to provide a multi-scale topology optimization method for anisotropic periodic structures / materials based on isogeometric analysis that can improve structural performance from both macroscopic and microscopic scales and from both structural and material perspectives while ensuring the macroscopic periodicity of the structure.

[0007] The technical solution adopted by this invention to solve its technical problem is: a multi-scale topology optimization method for anisotropic periodic structural materials based on isogeometry. This method mainly realizes the multi-scale topology optimization design of anisotropic composite materials through two stages: periodic material distribution optimization and periodic structure / material multi-scale parallel topology optimization. The purpose of the periodic material distribution optimization stage is to determine the distribution position of various microstructures in the macroscopic periodic structure and to determine the volume fraction of various microstructures in the periodic structure / material multi-scale parallel topology optimization stage. The purpose of the periodic structure / material multi-scale parallel topology optimization stage is to determine the macroscopic periodic structure, various microstructures, and the corresponding equivalent elastic matrix.

[0008] The specific implementation steps of the technical solution described in this invention are as follows:

[0009] (1) Based on the geometric characteristics of the structure in the actual engineering, determine the design domain of the macro-periodic structure using isogeometric analysis. Construct the control points and element information of the macro-periodic structure using NURBS spline surfaces in the isogeometric analysis method, and calculate the Gaussian point information and IGA basis function information of the macro-periodic structure. Based on the performance requirements of the structure in the actual engineering, determine the volume constraints of the macro-periodic structure, the initial relative density of the control points of the macro-periodic structure, and input the principal and secondary Poisson ratios and elastic moduli of the anisotropic material, as well as material orientation angles and other material properties. Divide the macro-periodic structure design domain into Ms design subdomains, and classify the control points and element information of the macro-periodic structure design domain according to the subdomain division scheme. The division scheme is as follows:

[0010] Ms=Mx×My (1)

[0011] Ns=Nx×Ny (2)

[0012] In the formula, Mx and My are the number of design subdomains in the x and y directions within the design domain, respectively, and Ns, Nx, and Ny represent the total number of control points in the design subdomain and the number of control points in the x and y directions, respectively.

[0013] (2) Construct the natural coordinate system (x, y) and the material coordinate system in the anisotropic material. The coordinate transformation relationship between the two coordinate systems enables rational transformation of material property parameters between the two coordinate systems, including the elastic matrix in the natural coordinate system. The expression is:

[0014]

[0015] In the formula, and These are the coordinate transformation matrix for anisotropic materials and the elastic matrix in the material coordinate system, respectively, where E1, E2, and ν are the values ​​of E1, E2, and ν. 12 and ν 21 In the material coordinate system and The tensile and compressive elastic moduli and Poisson's ratio in the direction satisfy the following relationship: G 12 Let ξ be the shear modulus, θ be the angle between the natural coordinate system and the material coordinate system, and define the ratio of Poisson's ratio in the directions ξ and η. (3) Optimize the periodic material distribution: (Poisson's ratio factor)

[0016] (I) Input the control points, Gaussian points, elements, IGA basis functions, initial relative density of control points, periodic design subdomain partitioning scheme, anisotropic material properties and coordinate transformation relationships of the initial design domain of the macroscopic periodic structure determined by steps (1) and (2); input the boundary conditions and the iteration termination conditions for periodic material distribution optimization.

[0017] (II) Calculation of displacement field of macroscopic periodic structure based on isogeometric analysis: (a) Solve the structural stiffness matrix of anisotropic material based on isogeometric analysis theory and SIMP material interpolation model, and set the penalty factor pe = 1 in this process; (b) Apply displacement boundary conditions and force load boundary conditions at control points related to external loads; (c) Establish discrete control equations and solve for the displacement parameter values ​​of control points in the design domain of macroscopic periodic structure; (d) Output the displacement vector U and the overall force load vector F of the control points in the design domain of macroscopic periodic structure.

[0018] (III) Establish a topology optimization mathematical model based on isogeometric analysis, with structural flexibility as the objective function. Its expression is:

[0019]

[0020] In the formula, C represents the structural flexibility, ρ represents the relative density vector at the control points, Ne represents the number of elements in the design domain of the macroscopic periodic structure, and K represents the overall stiffness matrix. ρ is the initial area of ​​the element. min =0.001 represents the minimum relative density at the control point, and V0 and V represent the structural volume before and after optimization, respectively. ρ is the specified volume fraction. i,j The relative density of the j-th control point within the i-th design subdomain of the macro-periodic structure design domain; These are IGA basis functions that use the coordinates of the cell center as the calculation point.

[0021] (IV) The sensitivity of the structural compliance objective function and volume constraint function of the structural topology optimization model is solved using the adjoint analysis method:

[0022]

[0023]

[0024] In the formula, These are the IGA basis functions with Gaussian points as the computation points;

[0025] (V) Periodic constraints are applied by redistributing the relative density and objective function sensitivity of control points with the same number within each design subdomain. The expression for this is:

[0026]

[0027]

[0028] In the formula, This represents the relative density of the redistribution of control points with the same number within each design subdomain. This represents the sensitivity of the objective function for the redistribution of control points with the same number within each design subdomain.

[0029] (VI) Write a program based on the Optimal Criteria (OC) method and update the design variables: input the relative density of the current control points after redistribution and the sensitivity of the objective function, update the relative density of the control points according to the OC method and calculate the total volume of the updated design domain, set a new interpolation point based on the difference in total volume before and after the update to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points.

[0030] (VII) Calculate the relative density difference of each control point when inputting and outputting in (VI), and find the maximum relative density change value. Compare the maximum change value with the total loop iteration termination condition set in (I) to determine whether the termination condition is met. If the termination condition is not met, feed back the relative density of the control point output in (VI) to (II) for re-iteration. If the iteration termination condition is met, the iteration terminates and the final control point relative density vector is output.

[0031] (VIII) The relative density values ​​at the center points of each element are obtained by interpolation based on the IGA basis functions and the relative density vectors of the control points. The relative density values ​​at the center points of each element are then used as the relative density values ​​of that element. The obtained relative density vectors of the elements are regularized to determine the distribution positions of various microstructures in the macroscopic periodic structure and the volume fractions of various microstructures. The regularization calculation formula is as follows:

[0032]

[0033] In the formula, Let be the relative density of the i-th unit within the ζ-th type of microstructure region in the macro-periodic structure; and These are the upper and lower boundaries of the relative density of units within the ζ-th type of microstructure region, respectively. The total number of units within the ζ-th type of microstructure region of the macro-periodic structure; The relative density of all units within the ζ-th microstructure region after regularization is given.

[0034] The specific steps are as follows: (a) Output a continuous relative density cloud map of the macroscopic periodic structure, observe the clustering of the relative density of the units in the cloud map, and divide it into several non-overlapping intervals, each interval representing a microstructure; (b) Sum the unit densities in each interval and then average them, and use the average relative density as the relative density value of the units in that interval and the volume fraction of the microstructure represented by that interval; (c) Output the segmented relative density of the macroscopic periodic structure after equal division, which is the distribution position of various microstructures in the macroscopic periodic structure;

[0035] (4) Perform multi-scale parallel topology optimization of periodic structures / materials:

[0036] (I) Input the control points, Gaussian points, elements, IGA basis functions, periodic design subdomain partitioning scheme, relative density vector of control points, anisotropic material properties and coordinate transformation relationships, distribution positions of various microstructures in the macro-periodic structure, volume fraction of various microstructures, boundary conditions, and iterative termination conditions of multi-scale parallel topology optimization of periodic structure / material, determined by steps (1), (2) and (3).

[0037] (II) Based on the types of microstructures determined in (I), determine the shape of the initial design domain for each type of microstructure, construct the control points and element information of the initial design domain for each type of microstructure using NURBS spline surfaces in the isogeometric analysis method, calculate the Gaussian point information and IGA basis function information of the initial design domain for each type of microstructure, and determine the initial relative density of the control points of the design domain for each type of microstructure.

[0038] (III) Based on isogeometric analysis theory and SIMP material interpolation model, material property interpolation models for various microstructures are constructed; firstly, the relative density field of the design domain of macroscopic periodic structures and various microstructures is constructed, and its expression is:

[0039]

[0040]

[0041] In the formula, and These represent the relative densities of arbitrary calculation points and control points within the macro-periodic structural design domain element, respectively. Design domain NURBS basis functions for macroscopic periodic structures; and Let be the relative densities of arbitrary computation points and control points within the design domain element of the ζ-th microstructure, respectively. Let M and m be the NURBS basis functions for the design domain of the ζ-th type of microstructure; M and m be the identifiers of the macroscopic periodic structure and the microstructure, respectively; then the material property interpolation models for various microstructures based on isogeometric analysis theory and SIMP material interpolation model can be expressed as:

[0042]

[0043] In the formula, The elasticity matrix after applying a penalty to the design domain of the ζ-th type of microstructure, where pk is the penalty factor for the relative density of microstructure control points;

[0044] (IV) Based on isogeometric analysis theory, SIMP material interpolation model, and energy homogenization method, the equivalent elastic matrix of various microstructures is calculated, and its expression is:

[0045]

[0046] In the formula, Let be the equivalent elasticity matrix of the ζ-th type microstructure. and These represent the displacement fields within the ζ-th type of microstructure before and after applying the test strain. The area of ​​the design domain for the ζth type of microstructure, where Θ is the number of microstructure types; For linearly independent element test strain fields within the ζ-th type microstructure, The unknown strain field within the ζ-th type of microstructure;

[0047] (V) Calculate the displacement field of a macroscopic periodic structure based on isogeometric analysis. The specific steps are as follows:

[0048] (a) Solving the stiffness matrix of anisotropic material structures considering multiple microstructures based on isogeometric analysis theory and SIMP material interpolation model; First, constructing a macroscopic anisotropic multi-scale material interpolation model considering multiple microstructures based on SIMP material interpolation model:

[0049]

[0050] In the formula, D M Let pe be the overall elasticity matrix after applying a penalty to the macro-cyclical structure, where pe is the penalty factor. Let be the relative density at any calculated point in the region containing the ζ-th type of microstructure in the macroscopic periodic structure; then, based on the isogeometric analysis theory and the SIMP material interpolation model, the structural stiffness matrix of the anisotropic material is:

[0051]

[0052] In the formula, Ω M and These are the macroscopic periodic structure physical domain and the parent space, respectively; their mapping relationship is shown in [reference needed]. Figure 3 J ξη and These are the Jacobian transformation matrices for mapping the physical domain to the parameter domain and the parameter domain to the parent space, respectively.

[0053] (b) Apply displacement boundary conditions and force load boundary conditions at control points associated with external loads;

[0054] (c) Establish discrete control equations and solve for the displacement parameter values ​​of control points within the design domain of the macroscopic periodic structure;

[0055] (d) Output the displacement vector U and the overall force load vector F of the control points in the macroscopic periodic structural design domain;

[0056] (VI) Establish a multi-scale topology optimization mathematical model for periodic structures / materials based on isogeometric analysis, with structural flexibility as the objective function and the relative densities of control points in the macro-periodic structural design domain and various micro-structural design domains as design variables. Its expression is:

[0057]

[0058] In the formula, J is the objective function related to macroscopic design variables and various microstructural design variables; ρ M and ρ m These represent the relative density vectors of control points in the macroscopic periodic structure design domain and control points in various microstructure design domains, respectively. and These represent the relative density vector and displacement vector of the control points in the region where the ζ-th type of microstructure is located in the macro-periodic structural design domain, respectively. U represents the relative density vector of the control points in the design domain of the ζ-th type of microstructure; M and F M Represents the overall displacement vector and load vector of a macroscopic periodic structure; Let be the number of grids in the design domain for the ζ-th type of microstructure; The total number of elements in the design domain of the ζ-th type of microstructure within the macro-periodic structure; Let be the initial area of ​​the unit cell in the design domain of the ζ-th type of microstructure within the macro-periodic structure; Let be the initial area of ​​the unit cell of the ζ-th type of microstructure; and τ ζ V represents the material volume fractions specified for the macrostructure and the ζ-type microstructure, respectively; M and These represent the volumes before and after macroscopic structural optimization, respectively. and These represent the volumes before and after optimization of the ζ-th type of microstructure; This represents the relative density of the j-th control point within the i-th design subdomain of the macrostructure; ρ represents the relative density of the i-th control point within the ζ-th microstructure; min =0.001 represents the minimum relative density of the control points; It is an IGA basis function in a macroscopic periodic structure that uses the coordinates of the unit center as the calculation point; For the IGA basis functions in the design domain of the ζ-th type of microstructure, the coordinates of the element center are used as the calculation points; The relative density of control points for the domain unit of the ζ-th type of microstructure within the macrostructure; Design the relative density of domain unit control points for the ζ-th type of microstructure;

[0059] (VII) The sensitivity of the structural compliance objective function and volume constraint function of the mathematical model for multi-scale topology optimization of periodic structures / materials to macroscopic design variables is determined using the adjoint analysis method:

[0060]

[0061]

[0062] In the formula, The relative density of control points in the region where the ζ-type microstructure is located within the macro-periodic structure design domain;

[0063] (VIII) The sensitivity of the structural compliance objective function and microstructure volume constraint function of the mathematical model for multi-scale topology optimization of periodic structures / materials to microstructure design variables is determined using the adjoint analysis method:

[0064]

[0065]

[0066] In the formula, The first-order partial derivative of the homogenized elastic tensor with respect to the microscopic design variables, obtained from the topological calculation of the ζ-th type of microstructure, is expressed as follows:

[0067]

[0068] In the formula, Let IGA be the basis function matrix of the computation points in the design domain of the ζ-th type of microstructure;

[0069] (IX) Periodic constraints are imposed by redistributing the relative density and objective function sensitivity of control points with the same number within each design subdomain of the macro-periodic structure. The expression for this redistribution is as follows:

[0070]

[0071]

[0072] In the formula, This represents the relative density of the redistribution of control points with the same number within each design subdomain of a macroscopic periodic structure. The sensitivity of the objective function represents the redistribution of control points with the same number within each design subdomain of a macro-periodic structure.

[0073] (X) Write a program based on the Optimal Criteria (OC) method and update the macro-periodic structure and various micro-structure design variables. The specific steps are as follows: (a) Input the relative density and objective function sensitivity of the current macro-periodic structure control points after redistribution. Update the relative density of the control points according to the OC method and calculate the total volume of the updated design domain. Use the difference between the total volume of the macro-periodic structure before and after the update to set a new interpolation point to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points; (b) Input the relative density and objective function sensitivity of the first type of micro-structure control points. Update the relative density of the control points according to the OC method and calculate the total volume of the updated micro-structure design domain. Use the difference between the total volume before and after the update to set a new interpolation point to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points; (c) The update and iteration of other types of micro-structure design variables are the same as the update and iteration of the first type of micro-structure design variables.

[0074] (XI) Calculate the relative density difference of each macro-periodic structure control point in the input and output of (X), and find the maximum relative density change value. Compare the maximum change value with the total loop iteration termination condition set in (I) to determine whether the termination condition is met. If the termination condition is not met, feed back the relative density of the macro-periodic structure design domain control point output in (X) to (III) for re-iteration. If the iteration termination condition is met, the iteration terminates and the final relative density vector of the macro-periodic structure design domain and various micro-structure design domain control points is output.

[0075] (XII) Output the optimal macroscopic periodic structure and various microstructure topologies based on isogeometric analysis, as well as the corresponding equivalent elasticity matrix.

[0076] The beneficial effects of this invention are as follows: The isogeometric analysis method adopted in this invention achieves the unification of the geometric model and the analysis model by using the spline function describing the geometric shape as the shape function of the solution domain for analysis and calculation. Moreover, the high-order continuous spline function and the accurate geometric description also ensure the accuracy of the analysis and calculation. The periodic structure / material multi-scale topology optimization design of anisotropic composite materials is realized through two stages: periodic material distribution optimization and periodic structure / material multi-scale parallel topology optimization. This can improve the product structural performance from both macroscopic and microscopic scales and from both structural and material perspectives while ensuring the macroscopic periodicity of the structure. It not only overcomes the defects of non-optimal performance of most current periodic porous composite material structures that rely on experience design, but also provides a clear and simple method for designing anisotropic periodic structure / material multi-scale structures. It can be closely integrated with engineering practice and has great theoretical research and engineering application value. Attached Figure Description

[0077] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0078] Figure 1 This is a flowchart illustrating the programming and calculation of the multi-scale topology optimization method for anisotropic periodic structural materials based on isogeometry described in this invention.

[0079] Figure 2 This refers to the mapping relationship between the physical domain, the parameter domain, and the parent space in the isogeometric analysis method described in this invention.

[0080] Figure 3 This is the macroscopic periodic structure design subdomain partitioning scheme described in this invention.

[0081] Figure 4 The orthogonal anisotropic material coordinate system and natural coordinate system described in this invention

[0082] Figure 5 This is a schematic diagram of the macroscopic periodic structure design domain according to an embodiment of the present invention.

[0083] Figure 6 This refers to the density distribution using the variable thickness method when the Poisson's ratio factor Bt = 0.6 in the embodiments described in this invention.

[0084] Figure 7 This refers to the density distribution using the variable thickness method when the Poisson's ratio factor Bt = 1 in the embodiments described in this invention.

[0085] Figure 8 This refers to the density distribution using the variable thickness method when the Poisson's ratio factor Bt = 1.5 in the embodiments described in this invention.

[0086] Figure 9 This refers to the density distribution using the variable thickness method when the Poisson's ratio factor Bt = 2 in the embodiments described in this invention.

[0087] Figure 10 This refers to the regularized block density distribution with a Poisson ratio factor Bt = 0.6 in the embodiments described in this invention.

[0088] Figure 11 This is the regularized block density distribution with Poisson ratio factor Bt = 1 in the embodiments described in this invention.

[0089] Figure 12 This is the regularized block density distribution with Poisson ratio Bt = 1.5 in the embodiments of the present invention.

[0090] Figure 13 This is the regularized block density distribution with Poisson ratio factor Bt = 2 in the embodiments described in this invention.

[0091] Figure 14 These are schematic diagrams of the initial design domains of various microstructures according to embodiments of the present invention.

[0092] Figure 15 This refers to the block-type periodic structure, various microstructures, and corresponding equivalent elastic matrix when the Poisson's ratio factor is 0.6 in the embodiments described in this invention.

[0093] Figure 16 This refers to the block-type periodic structure, various microstructures, and corresponding equivalent elastic matrix when the Poisson's ratio factor is 1 in the embodiments described in this invention.

[0094] Figure 17 This refers to the block-type periodic structure, various microstructures, and corresponding equivalent elastic matrix when the Poisson's ratio factor is 1.5 in the embodiments described in this invention.

[0095] Figure 18 This refers to the block-type periodic structure, various microstructures, and corresponding equivalent elastic matrix when the Poisson's ratio factor is 2 in the embodiments described in this invention. Detailed Implementation

[0096] Figure 1 This is a flowchart illustrating the programming and calculation of the multi-scale topology optimization method for isotropic anisotropic periodic structural materials based on the present invention. (See attached flowchart.) Figure 1 The specific process is as follows:

[0097] (1) Based on the geometric characteristics of the structure in the actual engineering, the design domain of the macroscopic periodic structure of the isogeometric analysis is determined. The control points and element information of the macroscopic periodic structure are constructed using NURBS spline surfaces in the isogeometric analysis method. The Gaussian point information and IGA basis function information of the macroscopic periodic structure are calculated. The formula of the IGA basis function is as follows:

[0098]

[0099]

[0100]

[0101] Where ξ is the coordinate variable of any calculation point in the parameter space, ξ i+1 Let N be the coordinate variable of the (i+1)th node in the parameter space, p and q be the orders of the isogeometric basis functions, u and v represent the coordinate variables of any computation point in the ξ and η directions in the parameter space, respectively. i,p (u) represents the one-dimensional B-spline curve value of the i-th control point with respect to the u-coordinate, N j,q (v) represents the one-dimensional B-spline curve value of the j-th node on the v coordinate. Control point P i,j The IGA basis function value at ω i,j For non-uniform B-spline curves at control point P i,j Weight at each location;

[0102] Based on the performance requirements of the structure in actual engineering, determine the volume constraints of the macro-periodic structure, the initial relative density of the control points of the macro-periodic structure, and input the principal and secondary Poisson's ratios and elastic moduli of the anisotropic material, as well as material orientation angles and other material properties; see [link / reference]. Figure 2 The macro-periodic structural design domain is divided into Ms design subdomains, and the control points and element information of the macro-periodic structural design domain are classified according to the subdomain division scheme. The division scheme is as follows:

[0103] Ms=Mx×My (4)

[0104] Ns=Nx×Ny (5)

[0105] In the formula, Mx and My are the number of design subdomains in the x and y directions within the design domain, respectively, and Ns, Nx, and Ny represent the total number of control points in the design subdomain and the number of control points in the x and y directions, respectively.

[0106] (2) Constructing such Figure 3 The natural coordinate system (x, y) and material coordinate system in the anisotropic material are shown. The coordinate transformation relationship between the two coordinate systems enables rational transformation of material property parameters between the two coordinate systems, including the elastic matrix in the natural coordinate system. The expression is:

[0107]

[0108] In the formula, and These are the coordinate transformation matrix for anisotropic materials and the elastic matrix in the material coordinate system, respectively, where E1, E2, and ν are the values ​​of E1, E2, and ν. 12 and ν 21In the material coordinate system and The tensile and compressive elastic moduli and Poisson's ratio in the direction satisfy the following relationship: G 12 Let ξ be the shear modulus, θ be the angle between the natural coordinate system and the material coordinate system, and define the ratio of Poisson's ratio in the directions ξ and η. (3) Optimize the periodic material distribution: (Poisson's ratio factor)

[0109] (3.1) Input the control points, Gaussian points, elements, IGA basis functions, initial relative density of control points, periodic design subdomain partitioning scheme, anisotropic material properties and coordinate transformation relationships of the initial design domain of the macroscopic periodic structure determined by steps (1) and (2); input the boundary conditions and the iteration termination conditions for periodic material distribution optimization.

[0110] (3.2) The displacement field of the macroscopic periodic structure is calculated based on isogeometric analysis. The specific steps are as follows:

[0111] (a) Solving the structural stiffness matrix of anisotropic materials based on isogeometric analysis theory and SIMP material interpolation model; First, using the relative density of control points as the design variable, and employing a quadratic NURBS spline function with high-order continuity as the shape function of the analysis and calculation solution domain, each element in the discrete design domain is determined by the information of (p+1)×(q+1) control points. The relative density of any point within any element can be obtained by interpolating the relative densities of the (p+1)×(q+1) control points of this element and the corresponding NURBS basis function, as expressed below:

[0112]

[0113] In the formula, ρ g Let ρ be the relative density at any calculation point within the element. I To control the relative density, The first function is the NURBS basis function. Secondly, based on the SIMP material interpolation model, a hypothetical material with a variable relative density between 0 and 1 is introduced. The expression for the elastic modulus based on the SIMP model is then as follows:

[0114]

[0115] In the formula, pe is the penalty factor, E0 is the elastic modulus of a given solid material, and ρ g (x) represents the relative density at any calculation point; finally, the structural stiffness matrix of the anisotropic material is constructed based on the isogeometric analysis theory and the SIMP material interpolation model:

[0116]

[0117] In the formula, Ω and These are the physical domain and the parent space, respectively; their mapping relationship can be found in [reference needed]. Figure 3 J ξη and These are the Jacobian transformation matrices for mapping the physical domain to the parameter domain and the parameter domain to the parent space, respectively.

[0118] (b) Apply displacement boundary conditions and force load boundary conditions at control points associated with external loads;

[0119] (c) Establish discrete control equations and solve for the displacement parameter values ​​of control points within the design domain of the macroscopic periodic structure;

[0120] (d) Output the displacement vector U and the overall force load vector F of the control points in the macroscopic periodic structural design domain;

[0121] (3.3) Establish a topology optimization mathematical model based on isogeometric analysis with structural flexibility as the objective function. Its expression is:

[0122]

[0123] In the formula, C represents the structural flexibility, ρ represents the relative density vector of the control points, and Ne represents the number of elements in the design domain of the macroscopic periodic structure. ρ is the initial area of ​​the element. min =0.001 represents the minimum relative density at the control point, and V0 and V represent the structural volume before and after optimization, respectively. ρ is the specified volume fraction. i,j The relative density of the j-th control point within the i-th design subdomain of the macro-periodic structure design domain; These are IGA basis functions that use the coordinates of the cell center as the calculation point.

[0124] (3.4) The sensitivity of the structural compliance objective function and volume constraint function of the structural topology optimization model is solved using the adjoint analysis method:

[0125]

[0126]

[0127] In the formula, These are the IGA basis functions with Gaussian points as the computation points;

[0128] (3.5) Periodic constraints are applied by redistributing the relative density and objective function sensitivity of control points with the same number within each design subdomain, as expressed in the following expression:

[0129]

[0130]

[0131] In the formula, This represents the relative density of the redistribution of control points with the same number within each design subdomain. This represents the sensitivity of the objective function for the redistribution of control points with the same number within each design subdomain.

[0132] (3.6) Write a program based on the Optimal Criteria (OC) method and update the design variables: Input the relative density of the current control points after redistribution and the sensitivity of the objective function. Update the relative density of the control points according to the OC method and calculate the total volume of the updated design domain. Use the difference in total volume before and after the update to set a new interpolation point to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points. The optimization criterion method is performed according to the following relationship:

[0133]

[0134] In the formula, k is the iteration step, κ and β are the movement limit and damping coefficient, respectively; The optimization criterion is expressed as follows:

[0135]

[0136] In the formula, δ1 is the Lagrange multiplier determined by the bisection method;

[0137] (3.7) Calculate the relative density difference of each control point at the input and output in (3.6), and find the maximum relative density change value. Compare the maximum change value with the total loop iteration termination condition set in (3.1) to determine whether the termination condition is met. If the termination condition is not met, feed back the relative density of the control points output in (3.6) to (3.2) for re-iteration. If the iteration termination condition is met, the iteration terminates and the final control point relative density vector is output. The convergence condition is based on the following relationship:

[0138]

[0139] In the formula, and These are the maximum relative density changes at step (k+1) and step (k), respectively, and ε is the iteration termination condition preset in step (3.2).

[0140] (3.8) Calculate the relative density of the elements and perform regularization to determine the distribution location of various microstructures in the macroscopic periodic structure and the volume fraction of various microstructures; first, based on the IGA basis function and the relative density vector of the control points, the relative density value at the center point of each element is obtained, and the relative density value at the center point of the element is used as the relative density value of the element. The relative density of the elements is solved according to the following relationship:

[0141]

[0142] Secondly, the obtained unit relative density vector is regularized to determine the distribution location of various microstructures in the macroscopic periodic structure and the volume fraction of various microstructures. The regularization calculation formula is as follows:

[0143]

[0144] In the formula, Let be the relative density of the i-th unit within the ζ-th type of microstructure region in the macro-periodic structure; and These are the upper and lower boundaries of the relative density of units within the ζ-th type of microstructure region, respectively. The total number of units within the ζ-th type of microstructure region of the macro-periodic structure; The relative density of all units within the ζ-th microstructure region after regularization is given.

[0145] The specific steps are as follows: (a) Output the continuous relative density cloud map of the macroscopic periodic structure, observe the aggregation of the relative density of the units in the cloud map, and divide it into several non-overlapping intervals, each interval representing a microstructure; (b) Sum the unit densities in each interval and then average them, and use the average relative density as the relative density value of the units in that interval and the volume fraction of the microstructure represented by that interval; (c) Output the segmented relative density of the macroscopic periodic structure and the relative density value of the units after equal division, which is the distribution position of various microstructures in the macroscopic periodic structure and the volume fraction of various microstructures.

[0146] (4) Perform multi-scale parallel topology optimization of periodic structures / materials:

[0147] (4.1) Input the control points, Gaussian points, elements, IGA basis functions, periodic design subdomain partitioning scheme, relative density vector of control points, anisotropic material properties and coordinate transformation relationships, distribution positions of various microstructures in the macro-periodic structure, volume fractions of various microstructures and boundary conditions of the initial design domain of the macro-periodic structure determined by steps (1), (2) and (3); determine the iterative termination condition of the multi-scale parallel topology optimization of the periodic structure / material;

[0148] (4.2) Based on the microstructure types determined in (4.1), determine the shape of the initial design domain for each type of microstructure. Use NURBS spline surfaces in the isogeometric analysis method to construct the control points and element information of the initial design domain for each type of microstructure. Calculate the Gaussian point information and IGA basis function information of the initial design domain for each type of microstructure to determine the initial relative density of the control points of each type of microstructure design domain. (4.3) Based on the isogeometric analysis theory and the SIMP material interpolation model, construct the material property interpolation model for each type of microstructure. First, construct the relative density field of the macroscopic periodic structure and the design domain for each type of microstructure. Its expression is:

[0149]

[0150]

[0151] In the formula, and These represent the relative densities of arbitrary calculation points and control points within the macro-periodic structural design domain element, respectively. Design domain NURBS basis functions for macroscopic periodic structures; and Let be the relative densities of arbitrary computation points and control points within the design domain element of the ζ-th microstructure, respectively. Let M and m be the NURBS basis functions for the design domain of the ζ-th type of microstructure; M and m be the identifiers of the macroscopic periodic structure and the microstructure, respectively; then the material property interpolation models for various microstructures based on isogeometric analysis theory and SIMP material interpolation model can be expressed as:

[0152]

[0153] In the formula, The elasticity matrix after applying a penalty to the design domain of the ζ-th type of microstructure, where pk is the penalty factor for the relative density of microstructure control points;

[0154] (4.4) Based on the isogeometric analysis theory, SIMP material interpolation model, and energy homogenization method, the equivalent elastic matrix of various microstructures is calculated, and its expression is:

[0155]

[0156] In the formula, Let be the equivalent elasticity matrix of the ζ-th type microstructure. and These represent the displacement fields within the ζ-th type of microstructure before and after applying the test strain. The area of ​​the design domain for the ζth type of microstructure, where Θ is the number of microstructure types; For linearly independent element test strain fields within the ζ-th type microstructure, The unknown strain field within the ζ-th type of microstructure;

[0157] (4.5) The displacement field of the macroscopic periodic structure is calculated based on isogeometric analysis. The specific steps are as follows:

[0158] (a) Solving the stiffness matrix of anisotropic material structures considering multiple microstructures based on isogeometric analysis theory and SIMP material interpolation model; First, constructing an anisotropic multi-scale material interpolation model considering multiple microstructures based on SIMP material interpolation model:

[0159]

[0160] In the formula, D M Let pe be the overall elasticity matrix after applying a penalty to the macro-cyclical structure, where pe is the penalty factor. Let be the relative density at any calculated point in the region containing the ζ-th type of microstructure in the macroscopic periodic structure; then, based on the isogeometric analysis theory and the SIMP material interpolation model, the structural stiffness matrix of the anisotropic material is:

[0161]

[0162] In the formula, Ω M and These are the macroscopic periodic structure physical domain and the parent space, respectively; their mapping relationship is shown in [reference needed]. Figure 3 J ξη and These are the Jacobian transformation matrices for mapping the physical domain to the parameter domain and the parameter domain to the parent space, respectively.

[0163] (b) Apply displacement boundary conditions and force load boundary conditions at control points associated with external loads;

[0164] (c) Establish discrete control equations and solve for the displacement parameter values ​​of control points within the design domain of the macroscopic periodic structure;

[0165] (d) Output the displacement vector U and the overall force load vector F of the control points in the macroscopic periodic structural design domain;

[0166] (4.6) Establish a multi-scale topology optimization mathematical model for periodic structures / materials based on isogeometric analysis, with structural flexibility as the objective function and the relative density of control points in the macro-periodic structural design domain and various micro-structural design domains as design variables. Its expression is:

[0167]

[0168] In the formula, J is the objective function related to macroscopic design variables and various microstructural design variables; ρ M and ρ mThese represent the relative density vectors of control points in the macroscopic periodic structure design domain and control points in various microstructure design domains, respectively. and These represent the relative density vector and displacement vector of the control points in the region where the ζ-th type of microstructure is located in the macro-periodic structural design domain, respectively. U represents the relative density vector of the control points in the design domain of the ζ-th type of microstructure; M and F M Represents the overall displacement vector and load vector of a macroscopic periodic structure. Let be the number of grids in the design domain for the ζ-th type of microstructure; Let be the initial area of ​​the unit cell in the design domain of the ζ-th type of microstructure within the macro-periodic structure; Let be the initial area of ​​the unit cell of the ζ-th type of microstructure; and τ ζ V represents the material volume fractions specified for the macrostructure and the ζ-type microstructure, respectively; M and These represent the volumes before and after macroscopic structural optimization, respectively. and These represent the volumes before and after optimization of the ζ-th type of microstructure; This represents the relative density of the j-th control point within the i-th design subdomain of the macrostructure; ρ represents the relative density of the i-th control point within the ζ-th microstructure; min =0.001 represents the minimum relative density of the control points; It is an IGA basis function in a macroscopic periodic structure that uses the coordinates of the unit center as the calculation point; For the IGA basis functions in the design domain of the ζ-th type of microstructure, the coordinates of the element center are used as the calculation points; The relative density of control points for the domain unit of the ζ-th type of microstructure within the macrostructure; Design the relative density of domain control points for the ζ-th type of microstructure;

[0169] (4.7) The sensitivity of the structural compliance objective function and volume constraint function of the mathematical model for multi-scale topology optimization of periodic structures / materials to macroscopic design variables is determined using the adjoint analysis method:

[0170]

[0171]

[0172] In the formula, The relative density of control points in the region where the ζ-type microstructure is located within the macro-periodic structure design domain;

[0173] (4.8) The sensitivity of the structural compliance objective function and microstructure volume constraint function of the mathematical model for multi-scale topology optimization of periodic structures / materials to microstructure design variables is determined by the adjoint analysis method:

[0174]

[0175]

[0176] In the formula, The first-order partial derivative of the homogenized elastic tensor with respect to the microscopic design variables, obtained from the topological calculation of the ζ-th type of microstructure, is expressed as follows:

[0177]

[0178] In the formula, Let IGA be the basis function matrix of the computation points in the design domain of the ζ-th type of microstructure;

[0179] (4.9) Periodic constraints are applied by redistributing the relative density and objective function sensitivity of control points with the same number within each design subdomain of the macro-periodic structure. The expression for this redistribution is as follows:

[0180]

[0181]

[0182] In the formula, This represents the relative density of the redistribution of control points with the same number within each design subdomain of a macroscopic periodic structure. The sensitivity of the objective function represents the redistribution of control points with the same number within each design subdomain of a macro-periodic structure.

[0183] (4.10) Write a program based on the Optimal Criteria (OC) method and update the design variables of the macro-periodic structure and various microstructures. The specific steps are as follows: (a) Input the relative density and objective function sensitivity of the current macro-periodic structure control points after redistribution. Update the relative density of the control points according to the OC method and calculate the total volume of the updated design domain. Set a new interpolation point based on the difference in the total volume of the macro-periodic structure before and after the update to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points. The macro-periodic structure optimization criterion method is performed according to the following relationship:

[0184]

[0185] In the formula, k is the iteration step, κ and β are the movement limit and damping coefficient, respectively; The optimization criterion is expressed as follows:

[0186]

[0187] In the formula, δ1 is the Lagrange multiplier determined by the bisection method;

[0188] (b) Input the relative density of the first type of microstructure control points and the sensitivity of the objective function. Update the relative density of the control points according to the OC method and calculate the total volume of the updated microstructure design domain. Use the difference in total volume before and after the update to set a new interpolation point to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points. The microstructure optimization criterion method is performed according to the following relationship:

[0189]

[0190] In the formula, k is the iteration step, κ and β are the movement limit and damping coefficient, respectively; The optimization criterion is expressed as follows:

[0191]

[0192] In the formula, λ1 is the Lagrange multiplier determined by the bisection method;

[0193] (c) The update and iteration methods for other types of microstructure design variables are the same as those for the first type of microstructure design variables;

[0194] (4.11) Calculate the relative density difference of each macroscopic periodic structure control point at the input and output in (4.9), and find the maximum relative density change value. Compare the maximum change value with the total loop iteration termination condition set in (4.1) to determine whether the termination condition is met. If the termination condition is not met, feed back the relative density of the macroscopic periodic structure design domain control points output in (4.9) to (4.3) for re-iteration. If the iteration termination condition is met, the iteration terminates and the final relative density vectors of the macroscopic periodic structure design domain and various microstructure design domain control points are output. The convergence condition is based on the following relationship:

[0195]

[0196] In the formula, and These are the maximum relative density changes at step (k+1) and step (k), respectively, and ε is the iteration termination condition preset in step (4.1).

[0197] (4.12) Output the optimal macroscopic periodic structure and various microstructure topologies based on isogeometric analysis, as well as the corresponding equivalent elasticity matrix.

[0198] The following is an example of the application of the method of the present invention in engineering practice:

[0199] See Figure 5 In this embodiment, the design domain is a square with a length and width of 100mm, and the principal elastic modulus of the material is E1 = 2.06 × 10⁻⁶ mm. 11 Pa, Poisson's ratio is ν 12 =0.3, material orientation angle θ = 0°; the left side of the design domain is fully constrained, and the center of the right side is subjected to a downward concentrated force F = 1000N; the volume constraint of the macroscopic periodic structure is 50%, the material penalty factor is 3, and the initial relative density of the control points is 0.5; the entire design domain is discretized by 120×120 control points and 118×118 elements; the design domain is divided into Ms = 3×3 design subdomains; the optimal anisotropic periodic multiscale structure with Poisson ratio Bt = 0.6, 1, 1.5, and 2 was calculated, and the corresponding optimal topology, regularized block density distribution, block periodic structure, various microstructures, and corresponding equivalent elastic matrix were obtained.

[0200] The specific implementation steps of this invention for this example are as follows:

[0201] (1) Based on the geometric characteristics of the structure in the actual engineering, determine the design domain of the macro-periodic structure of the isogeometric analysis, construct the control points and element information of the macro-periodic structure using NURBS spline surfaces in the isogeometric analysis method, and calculate the Gaussian point information and IGA basis function information of the macro-periodic structure; based on the performance requirements of the structure in the actual engineering, determine the volume constraint of the macro-periodic structure, the initial relative density of the control points of the macro-periodic structure, and input the principal and secondary Poisson ratios and elastic modulus of the anisotropic material, as well as the material orientation angle and other material properties; divide the macro-periodic structure design domain into Ms = 2 × 2 design subdomains, and classify the control points and element information of the macro-periodic structure design domain according to the design subdomain division scheme;

[0202] (2) Construct the natural coordinate system (x, y) and the material coordinate system in the anisotropic material. Determine the coordinate transformation relationship between them and solve for the elastic tensor matrix in the natural coordinate system;

[0203] (3) Perform periodic material distribution optimization:

[0204] (3.1) Input the control points, Gaussian points, elements, IGA basis functions, initial relative density of control points, periodic design subdomain partitioning scheme, anisotropic material properties and coordinate transformation relationships of the initial design domain of the macroscopic periodic structure determined by steps (1) and (2); input the boundary conditions; set the iteration termination condition, that is, the iteration ends when the maximum change in the relative density of the discrete points before and after the update is less than 0.001;

[0205] (3.2) Solve the structural stiffness matrix of anisotropic materials based on the isogeometric analysis theory and SIMP material interpolation model; apply displacement boundary conditions and force load boundary conditions at control points related to external loads; establish discrete control equations and solve for the displacement parameter values ​​of control points in the macro-periodic structural design domain; output the displacement vector U and the overall force load vector F of the control points in the macro-periodic structural design domain.

[0206] (3.3) Establish a topology optimization mathematical model based on isogeometric analysis with structural flexibility as the objective function, and substitute the displacement vector U of the macro-periodic structural design domain control point and the overall force load vector F obtained in (3.2) to calculate the structural flexibility;

[0207] (3.4) The sensitivity of the structural compliance objective function and volume constraint function of the structural topology optimization model is solved by using the adjoint analysis method; (3.5) Periodic constraints are applied by redistributing the relative density and objective function sensitivity of control points with the same number in each design subdomain;

[0208] (3.6) Update the design variables according to the optimization criteria (OC), calculate the total volume of the updated design domain based on the new relative density of the control points, set the new interpolation point based on the difference in total volume before and after the update to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points.

[0209] (3.7) Calculate the relative density difference of each control point when inputting and outputting in (3.6), and find the maximum relative density change value. Compare the maximum change value with the total loop iteration termination condition set in (3.1) to determine whether the termination condition is met. If the termination condition is not met, feed back the relative density of the control points output in (3.6) to (3.2) for re-iteration. If the iteration termination condition is met, the iteration terminates and the final control point relative density vector is output.

[0210] (3.8) Calculate the relative density of the elements. Based on the IGA basis functions and the relative density vector of the control points, interpolate to obtain the relative density value at the center point of each element, and use the relative density value at the center point of the element as the relative density value of that element. Figure 6 , Figure 7 , Figure 8 and Figure 9 The optimal topologies for the variable thickness method are listed sequentially for Bt = 0.6, 1, 1.5, and 2. The clustering of relative cell density in the contour plot is observed, and based on this, it is divided into four non-overlapping intervals, namely [ρ...]. min The intervals [0.5, 0.5), [0.5, 0.7), [0.7, 0.8), and [0.8, 1] represent different microstructures. The unit densities within each interval are summed and then averaged. The average relative density is used as the relative density value of the unit within that interval and the volume fraction of the microstructure represented by that interval. The output is the segmented relative density of the macroscopic periodic structure and the average relative density value of the unit. This represents the distribution location of various microstructures in the macroscopic periodic structure and the volume fraction of various microstructures. Figure 10 , Figure 11 , Figure 12 and Figure 13 The regularized block density distributions for Bt = 0.6, 1, 1.5, and 2 are listed in sequence.

[0211] (4) Perform multi-scale parallel topology optimization of periodic structures / materials:

[0212] (4.1) Input the control points, Gaussian points, elements, IGA basis functions, periodic design subdomain partitioning scheme, relative density vector of control points, anisotropic material properties and coordinate transformation relationships, distribution positions of various microstructures in the macro-periodic structure, volume fraction of various microstructures and boundary conditions of the initial design domain of the macro-periodic structure determined by steps (1), (2) and (3); set the iteration termination condition for the multi-scale parallel topology optimization of periodic structure / material, that is, the iteration ends when the maximum change in the relative density of discrete points before and after the update is less than 0.001;

[0213] (4.2) Based on the four types of microstructures identified in (4.1), determine the initial design domain shape for each type of microstructure as follows: Figure 14 As shown, the white part in the middle is a void material with a relative density of 0.001. The overall design domain of the microstructure contains 60×60 control points. The control points and element information of the initial design domain of various microstructures are constructed using NURBS spline surfaces in the isogeometric analysis method. The Gaussian point information and IGA basis function information of the initial design domain of various microstructures are calculated, and the initial relative density of the control points of various microstructure design domains is determined as their respective volume constraints.

[0214] (4.3) Construct material property interpolation models for various microstructures based on isogeometric analysis theory and SIMP material interpolation model;

[0215] (4.4) Based on the isogeometric analysis theory, SIMP material interpolation model and energy homogenization method, calculate the equivalent elastic matrix of various microstructures. At this time, the input test strain is a 3×3 identity matrix.

[0216] (4.5) The displacement field of the macroscopic periodic structure is calculated based on isogeometric analysis. First, an anisotropic multi-scale material interpolation model considering multiple microstructures is constructed based on the SIMP material interpolation model. Then, the structural stiffness matrix of the anisotropic material based on isogeometric analysis theory and SIMP material interpolation model is solved. Second, displacement boundary conditions and force load boundary conditions are applied to the control points related to external loads. Third, discrete control equations are established to solve the displacement parameter values ​​of the control points in the design domain of the macroscopic periodic structure. Finally, the displacement vector U and the overall force load vector F of the control points in the design domain of the macroscopic periodic structure are output.

[0217] (4.6) Establish a mathematical model for multi-scale topology optimization of periodic structures / materials based on isogeometric analysis, with structural flexibility as the objective function and the relative density of control points in the macro-periodic structural design domain and various micro-structural design domains as design variables, and calculate the structural flexibility.

[0218] (4.7) The sensitivity of the structural compliance objective function and volume constraint function of the mathematical model for multi-scale topology optimization of periodic structures / materials to macroscopic design variables is determined by adjoint analysis.

[0219] (4.8) The sensitivity of the structural compliance objective function and microstructure volume constraint function of the mathematical model for multi-scale topology optimization of periodic structures / materials to microstructure design variables is obtained by using the adjoint analysis method;

[0220] (4.9) Periodic constraints are imposed by redistributing the relative density and objective function sensitivity of control points with the same number in each design subdomain of the macro-periodic structure.

[0221] (4.10) Update the macro-periodic structural design variables according to the optimization criteria (OC), take the moving limit κ = 0.02 and the damping coefficient β = 0.5, calculate the total volume of the updated macro-periodic structural design domain according to the new relative density of the control points, and set a new interpolation point based on the difference in total volume before and after the update to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points.

[0222] (4.11) Update the various microstructure design variables according to the optimization criterion OC method, take the movement limit κ = 0.02 and the damping coefficient β = 0.5, calculate the total volume of the updated microstructure design domain according to the new relative density of the control points, and set the new interpolation point by the difference between the total volume before and after the update to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue the iteration according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points.

[0223] (4.12) Calculate the relative density difference of each macro-periodic structure control point when inputting and outputting in (4.9), and find the maximum relative density change value. Compare the maximum change value with the total loop iteration termination condition set in (4.1) to determine whether the termination condition is met. If the termination condition is not met, feed back the relative density of the macro-periodic structure design domain control point output in (4.9) to (4.3) for re-iteration. If the iteration termination condition is met, the iteration terminates and the final relative density vector of the macro-periodic structure design domain and various micro-structure design domain control points is output.

[0224] (4.13) Output the optimal macroscopic periodic structure and various microstructure topologies based on isogeometric analysis, along with the corresponding equivalent elasticity matrices. The block-type periodic structure, various microstructures, and their corresponding equivalent elasticity matrices for Bt = 0.6, 1, 1.5, 2 are as follows: Figure 15 , Figure 16 , Figure 17 and Figure 18 As shown.

[0225] Although the present invention has been described in detail with reference to this embodiment, the above description does not limit the scope of protection of the present invention. Any modifications and improvements made based on the concept of the present invention shall be considered within the scope of protection of the present invention.

Claims

1. A multi-scale topology optimization method for anisotropic periodic structural materials based on isogeometry, characterized in that... Includes the following steps: (1) Based on the geometric characteristics of the structure in the actual engineering, the design domain of the macroscopic periodic structure in isogeometric analysis is determined. The control points and element information of the macroscopic periodic structure are constructed using NURBS spline surfaces in the isogeometric analysis method. The Gaussian point information and IGA basis function information of the macroscopic periodic structure are calculated. The formula of the IGA basis function is: ,in for Rank Each node The values ​​of a one-dimensional B-spline curve on the coordinate system. , ,in, It is the coordinate variable of any calculation point in the parameter space. It is the first in the parameter space The coordinate variables of each node, and The order of the geometric basis functions is equal. and They represent the parameter space respectively and Calculate the coordinate variables of any point in any direction. For the first Control point pair The one-dimensional B-spline curve values ​​of the coordinates. For the first Each node The values ​​of a one-dimensional B-spline curve on the coordinate system. Control points IGA basis function values ​​at that location, For non-uniform B-spline curves at control points The weights at each point are determined; based on the performance requirements of the structure in actual engineering, the volume constraints of the macro-periodic structure and the initial relative density of the control points of the macro-periodic structure are determined; the principal and secondary Poisson's ratios and elastic moduli of the anisotropic material, as well as material orientation angles and other material properties, are input; the design domain of the macro-periodic structure is divided into... The design subdomains are divided into several subdomains, and the control points and element information of the macro-periodic structural design domains are categorized according to the subdomain division scheme. The division scheme is as follows: , ,in, and Within the design domain and Number of design subdomains in the direction, , and Representing the total number of control points within the design subdomain and and Number of control points for direction; (2) Constructing a natural coordinate system in anisotropic materials and material coordinate system The coordinate transformation relationship between the two coordinate systems enables rational transformation of material property parameters between the two coordinate systems, including the elastic matrix in the natural coordinate system. The expression is: ,in and These are the coordinate transformation matrix for anisotropic materials and the elastic matrix in the material coordinate system, respectively. , , and In the material coordinate system and The tensile and compressive elastic moduli and Poisson's ratio in the direction satisfy the following relationship: , Shear modulus The angle between the natural coordinate system and the material coordinate system is defined as follows: and The ratio of Poisson's ratio in different directions Poisson's ratio factor; (3) Perform periodic material distribution optimization; (4) Perform multi-scale parallel topology optimization of periodic structures / materials.

2. The multi-scale topology optimization method for anisotropic periodic structural materials based on isogeometry according to claim 1, characterized in that... Step (3) includes the following specific steps: (a) Input the control points, Gaussian points, elements, IGA basis functions, initial relative density of control points, periodic design subdomain partitioning scheme, anisotropic material properties and coordinate transformation relationships of the initial design domain of the macroscopic periodic structure determined by steps (1) and (2); input the boundary conditions; Set the iteration termination condition; (b) Solve the structural stiffness matrix of anisotropic materials based on isogeometric analysis theory and SIMP material interpolation model; apply displacement boundary conditions and force load boundary conditions at control points related to external loads; establish discrete control equations and solve for the displacement parameter values ​​of control points in the macroscopic periodic structure design domain; Output the displacement vector of the control point in the design domain of the macroscopic periodic structure. and total force load vector ; (c) Establish a topology optimization mathematical model based on isogeometric analysis with structural flexibility as the objective function: in, For structural flexibility, Let the relative density vector of the control points be... The number of units in the design domain for macroscopic periodic structures. The initial area of ​​the unit. To determine the minimum relative density at the control points, and These represent the structural volumes before and after optimization, respectively. For the specified volume fraction, For the design of macro-periodic structures, the first domain The first design subdomain The relative density of each control point; The IGA basis functions are calculated using the coordinates of the unit center point; the displacement vectors of the control points in the macroscopic periodic structural design domain are obtained by substituting them into (b). and total force load vector To calculate structural flexibility; (d) The sensitivity of the structural compliance objective function and volume constraint function of the structural topology optimization model is solved by adjoint analysis; (e) Periodic constraints are imposed by redistributing the relative density and objective function sensitivity of control points with the same number within each design subdomain; (f) Write a program based on the Optimal Criteria (OC) method and update the design variables: input the relative density of the current control points after redistribution and the sensitivity of the objective function, update the relative density of the control points according to the OC method and calculate the total volume of the updated design domain, set a new interpolation point based on the difference in total volume before and after the update to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points. (g) Calculate the relative density difference of each control point when inputting and outputting in (f), and find the maximum relative density change value. Compare the maximum change value with the total loop iteration termination condition set in (a) to determine whether the termination condition is met. If the termination condition is not met, feed back the relative density of the control point output in (f) to (b) for re-iteration. If the iteration termination condition is met, the iteration terminates and the final control point relative density vector is output. (h) Calculate the relative density of the elements. Based on the IGA basis functions and the relative density vector of the control points, interpolate the relative density value at the center point of each element to obtain the relative density value of that element. Use the relative density value at the center point of the element as the relative density value of that element. Perform regularization processing on the obtained element relative density vector to determine the distribution position of various microstructures in the macroscopic periodic structure and the volume fraction of various microstructures. The regularization processing formula is as follows: in, For the first in the macro-cyclical structure The relative density of the i-th unit within the microstructure-like region; and The first The upper and lower boundaries of the relative density of units within the microstructure-like region; For the first macro-cyclical structure The total number of units within a microstructure-like region; For the first The relative density of all units within the microstructure region after normalization; output the relative density of the segmented units of the macro-periodic structure after equal division and the relative density value of the units after equal division, which is the distribution position of various microstructures in the macro-periodic structure and the volume fraction of various microstructures.

3. The multi-scale topology optimization method for anisotropic periodic structural materials based on isogeometry according to claim 1, characterized in that... Step (4) includes the following specific steps: (a) Input the control points, Gaussian points, elements, IGA basis functions, periodic design subdomain partitioning scheme, relative density vector of control points, anisotropic material properties and coordinate transformation relationships, distribution positions of various microstructures in the macro-periodic structure, volume fraction of various microstructures and boundary conditions of the initial design domain of the macro-periodic structure determined by steps (1), (2) and (3). Define the iterative termination condition for multi-scale parallel topology optimization of periodic structures / materials; (b) Based on the microstructure types determined in (a), determine the shape of the initial design domain for each type of microstructure, construct the control points and element information of the initial design domain for each type of microstructure using NURBS spline surfaces in the isogeometric analysis method, calculate the Gaussian point information and IGA basis function information of the initial design domain for each type of microstructure, and determine the initial relative density of the control points of the design domain for each type of microstructure. (c) Construct material property interpolation models for various microstructures based on isogeometric analysis theory and SIMP material interpolation model; (d) Calculate the equivalent elastic matrix of various microstructures based on the isogeometric analysis theory, SIMP material interpolation model and energy homogenization method; (e) Calculate the displacement field of the macroscopic periodic structure based on isogeometric analysis, construct an anisotropic multi-scale material interpolation model considering multiple microstructures based on the SIMP material interpolation model; solve the structural stiffness matrix of the anisotropic material based on isogeometric analysis theory and SIMP material interpolation model; apply displacement boundary conditions and force load boundary conditions at control points related to external loads; establish discrete control equations and solve for the displacement parameter values ​​of control points within the design domain of the macroscopic periodic structure. Output the displacement vector of the control point in the design domain of the macroscopic periodic structure. and total force load vector ; (f) Establish a multi-scale topology optimization mathematical model for periodic structures / materials based on isogeometric analysis, with structural flexibility as the objective function and the relative density of control points in the macro-periodic structural design domain and various micro-structural design domains as design variables: in, Objective functions related to macroscopic design variables and various microstructural design variables; and These represent the relative density vectors of control points in the macroscopic periodic structure design domain and control points in various microstructure design domains, respectively. and These represent the first in the macro-cyclical structure design domain. The relative density vector and displacement vector of the control points in the region where the microstructure is located; Indicates the first Relative density vector of control points in a microstructure-like design domain; and Represents the overall displacement vector and load vector of a macroscopic periodic structure. For the first The number of grids in a microstructure-like design domain; and These represent the number of subdomains in the macro-cyclical structure design and the number of control points within each subdomain, respectively. Within the macro-cyclical structure Initial cell area of ​​a microstructure-like design domain; For the first Initial area of ​​the microstructure-like unit; and They are respectively macrostructure and the first The volume fraction of materials specified for microstructure-like structures; and These represent the volumes before and after macroscopic structural optimization, respectively. and The first Volume before and after microstructure optimization; Indicates the first within the macrostructure The first design subdomain The relative density of each control point; This represents the minimum relative density of the control points. It is an IGA basis function in a macroscopic periodic structure that uses the coordinates of the unit center as the calculation point; For the first IGA basis functions in a microstructure-like design domain that use the coordinates of the element center as the computation point; For the first in the macro structure Relative density of control points in microstructure-like design domain units; For the first Relative density of control points in microstructure-like design domains; (g) The sensitivity of the structural compliance objective function and volume constraint function of the mathematical model for multi-scale topology optimization of periodic structures / materials to macroscopic design variables is determined by adjoint analysis. (h) The sensitivity of the structural compliance objective function and microstructure volume constraint function of the mathematical model for multi-scale topology optimization of periodic structures / materials to microstructure design variables is determined by adjoint analysis. (i) Periodic constraints are imposed by redistributing the relative density and objective function sensitivity of control points with the same number within each design subdomain of the macro-periodic structure, as expressed in the following expression: In the formula, This represents the relative density of the redistribution of control points with the same number within each design subdomain of a macroscopic periodic structure. The sensitivity of the objective function represents the redistribution of control points with the same number within each design subdomain of a macro-periodic structure. (j) Update the macro-periodic structure design variables according to the optimization criteria (OC), calculate the total volume of the updated macro-periodic structure design domain based on the new relative density of the control points, set a new interpolation point based on the difference in total volume before and after the update to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points. (k) Update various microstructure design variables according to the optimization criterion OC method, calculate the total volume of the updated microstructure design domain according to the new relative density of the control points, set a new interpolation point based on the difference between the total volume before and after the update to determine whether the iteration terminates. If it does not terminate, use the updated relative density of the control points and continue iterating according to the OC method. If the iteration terminates, stop the calculation and output the updated relative density of the control points. (l) Calculate the relative density difference of each macro-periodic structure control point at the input and output in (j), and find the maximum relative density change value. Compare the maximum change value with the total loop iteration termination condition set in (a) to determine whether the termination condition is met. If the termination condition is not met, feed back the relative density of the macro-periodic structure design domain control point output in (j) to (c) for re-iteration. If the iteration termination condition is met, the iteration terminates and the final relative density vector of the macro-periodic structure design domain and various micro-structure design domain control points is output. (m) Outputs the optimal macroscopic periodic structure and various microstructure topologies based on isogeometric analysis, as well as the corresponding equivalent elasticity matrix.

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