A Multi-Scale Topology Optimization Design Method for Geometric-Material Asymmetric Sandwich Structures

Abstract

This invention relates to a multi-scale topology optimization design method for geometrically and materially asymmetric sandwich structures. It employs a variable-thickness topology optimization method to optimize the thickness and material selection of the sandwich structure panels, and a parametric level set topology optimization method to optimize the design of multi-material sandwich microstructures. The optimized multi-material sandwich microstructures have smooth structural boundaries and clear interfaces between material phases. The use of an alternating active phase algorithm reduces the design variables and constraints for upper and lower panel optimization and sandwich microstructure optimization, effectively improving computational efficiency. This invention achieves joint optimization of the thickness and materials of the upper and lower panels of geometrically and materially asymmetric sandwich structures, as well as the multi-material sandwich microstructure. It not only obtains the optimal thickness and optimal material selection for the panels but also the optimal topological configuration of the multi-material sandwich microstructure, fully utilizing the material potential and effectively improving the mechanical properties of the sandwich structure while reducing its weight.

Description

Technical Field

[0001] This invention relates to the field of sandwich structure design methods, specifically to a multi-scale topology optimization design method for geometrically and materially asymmetric sandwich structures. Background Technology

[0002] Sandwich structures are a type of high-performance lightweight structure composed of high-strength solid panels and high-porosity porous cores. They possess outstanding advantages such as high specific stiffness / strength, impact resistance, thermal insulation, vibration reduction and noise reduction, high stability, and strong design flexibility, making them extremely valuable in engineering fields such as aerospace, automotive, and shipbuilding. Currently, most sandwich structure designs are based on the symmetry assumption, meaning that the core configuration is designed under the condition that the upper and lower panels have equal thicknesses, and all panels and cores are designed using a single material. This results in the underutilization of the design space for sandwich structures, severely restricting the improvement of their mechanical properties. Multi-scale topology optimization is an advanced structural design method that can achieve synergistic optimization of microscopic material configuration and macroscopic structural topology. It fully considers the coupling effect between microscopic materials and macroscopic structures, making it highly suitable for the design of geometrically and materially asymmetrical sandwich structures. It can achieve optimal mechanical performance design of sandwich structures with minimal material usage or at the lowest cost.

[0003] For the optimization design of sandwich structures, some research has been conducted by relevant personnel in this field. For example, Reference 1: Sun Z., Li D., Zhang WS, Shi SS, Guo X. Topological optimization of biomimetics sandwich structures with hybrid core and CFRP face sheets[J]. Composites Science and Technology, 2017, 142: 79-90. This paper proposes a topological optimization design method for sandwich structures based on the basic structure method, which realizes the design of a sandwich structure with carbon fiber panels and a lattice-honeycomb hybrid core. The core of this method is to invent a new type of sandwich structure by designing a lattice-honeycomb hybrid core, without involving the optimization of the thickness of the upper and lower panels of the sandwich structure. Reference 2: "Chu S., Gao L., Xiao M., Li H. Design of sandwich panels with trusscores using explicit topology optimization. Composites Science and Technology, 2017, 142: 79-90." "Structures, 2019, 210:892-905" discloses a topology optimization method for sandwich structures based on the moving deformable component method. This method improves the mechanical properties of the sandwich structure by optimizing the thickness of the upper and lower panels and the lattice core configuration. However, this method does not further optimize the topology of the sandwich structure's core configuration to fully explore the design space and material potential. Reference 3: "Zhang Y., Zhang L., Ding Z., Gao L., Xiao M., Liao WHA multiscale topological design method of geometrically asymmetric porous sandwich structures for minimizing The paper "dynamiccompliance.Materials&Design, 2022, 214:110404" discloses a geometrically asymmetric topology optimization method for sandwich structures, which is used to achieve synergistic optimization of the thickness of the upper and lower panels and the topology of the core, thereby improving the mechanical properties of the sandwich structure. However, in this method, the upper and lower panels and the core of the sandwich structure are designed with only a single material, failing to achieve multi-material design for the upper and lower panels and the core, thus failing to further expand the design space of the sandwich structure and improve its mechanical properties. Summary of the Invention

[0004] The technical problem this invention aims to solve is to provide a multi-scale topology optimization design method for sandwich structures with geometric-material asymmetry, addressing the above-mentioned problems and requirements. This invention, while considering the coordinated optimization of the thickness of the upper and lower panels and the core sandwich configuration of the sandwich structure, simultaneously introduces various materials with different mechanical properties to design a novel sandwich structure with geometric and material asymmetry. This fully expands the design space of the sandwich structure, unleashes the potential of materials, and maximizes the mechanical performance of the sandwich structure.

[0005] To solve the above technical problems, the present invention adopts the following technical solution:

[0006] A multi-scale topology optimization design method for geometrically-material asymmetric sandwich structures includes the following steps:

[0007] Step 1: The sandwich structure to be optimized consists of a solid panel and a core layer composed of a periodically repeating porous core microstructure. The sandwich structure to be optimized is first discretized into several slices of equal thickness, and the relative density of each material in each slice is set. initial value, The value of is between (0,1), m is the sequence number of the sandwich layer, n is the sequence number of the material type, n=1,2,…,N,m=1,2,…,M,N is the total number of material types used in the sandwich structure optimization, and M is the total number of slices after the sandwich structure is discretized.

[0008] Step 2: Optimize the model using the variable thickness method to obtain the relative density of each material in each slice of the sandwich structure. The calculation results of the relative density of each slice have two possibilities: 1. The relative density of a certain material is 1, and the relative densities of other materials are all 0; 2. The relative densities of all materials are greater than or equal to 0 and less than 1.

[0009] The sliced ​​layer in case 1 is used as the solid surface layer, and a material with a relative density of 1 is directly selected as the material for the corresponding solid surface layer. The sliced ​​layer in case 2 is used as the core layer. The arrangement of the solid surface layers and core layers is obtained. The relative density value of each material is obtained by summing the relative densities of each material in all core layers and averaging the results. n is the material type number;

[0010] Step 3: Take the core layer as the object to be optimized, and discretize the core layer into several periodic microstructure sandwich units according to the thickness of the core layer. Use the relative density value of each material in the core layer as the volume ratio constraint for the topology optimization of the periodic microstructure sandwich unit. Use the parameterized level set topology optimization method based on radial basis function interpolation to perform topology optimization on the microstructure sandwich unit to obtain the optimal distribution of each material in the design domain of the microstructure sandwich unit. The combination of the optimal distribution of each material in the design domain of the microstructure sandwich unit is the optimal topology of the microstructure sandwich unit.

[0011] Step 4: Periodically repeat the optimal topological configuration of the microstructure sandwich unit cell obtained in the previous step in the middle core layer of the sandwich structure to obtain the optimized complete structure of the middle core layer. Arrange the solid surface layer and core layer according to the arrangement method obtained in Step 1, and determine the solid surface layer material according to the preset requirements to obtain the optimized complete sandwich structure.

[0012] Furthermore, in step 2, the specific calculation model of the variable thickness optimization method is as follows:

[0013] Find:

[0014] Min:

[0015] St:G(ρ 1 )=V(ρ 1 -f1V0≤0

[0016] …

[0017] G(ρ n )=V(ρ n )-f n V0≤0

[0018]

[0019]

[0020]

[0021] in, Let represent the relative density value of the nth phase material contained in the m-th slice layer of the sandwich structure. The relative density values ​​of all finite elements within a slice layer are kept consistent. C is the objective function, represents the structural compliance value of the sandwich structure, Ω represents the total design domain contained in the sandwich structure, and ε ij and ε kl Let i, j, k, l = 1, 2, ..., d, where d is the spatial dimension. It is the elastic tensor of the finite element in the m-th layer of the sandwich structure. Interpolation calculations are performed using the elastic tensor matrices of all materials within the m-th layer, where G represents the volume fraction constraint of various materials in the sandwich structure, and f... n V is the volume fraction of the given nth phase material. max This represents the maximum allowable volume fraction of the sandwich structure, u is the macroscopic displacement field, and v is the space within the allowable displacement. The macroscopic virtual displacement field, f is the volume force acting on the design domain Ω of the sandwich structure, τ is the traction force acting on the boundary Γ, and ρ is the volume force acting on the boundary Γ. max and ρmin These are the upper and lower boundaries of the preset design variable values. If the obtained values ​​are... The lower limit value ρ min Then it is determined It is 0.

[0022] Furthermore, the aforementioned The elastic tensor matrices of all materials within the m-th layer are calculated using the following interpolation model:

[0023]

[0024] Where E n It is the elastic tensor of the nth phase material.

[0025] Furthermore, in step 2, the alternating active phase algorithm is used to transform the original N+1 phase multi-material layout optimization problem into N(N+1) / 2 two-phase material layout optimization sub-problems. The original multi-phase material layout optimization problem includes N phase solid materials and 1 phase porous material. The optimization model for the two-phase material layout optimization sub-problems is as follows:

[0026] Find:

[0027] Min:

[0028] St:G(ρ a )=V(ρ a )-f a V0≤0

[0029]

[0030]

[0031] 0≤ρ min ≤ρ a ≤ρ max

[0032] Where a and b represent the a-th phase material and the b-th phase material in the two-phase material layout optimization subproblem, respectively, C ab It is the objective function of the two-phase material layout optimization subproblem, based on the element elasticity tensor. The interpolation model calculates the elastic tensor of a finite element containing material phases a and b. G(ρ a The volume constraint of material phase a is given by ( ). In this two-phase material layout optimization subproblem, the design variable is the relative density ρ of material phase a. a The relative density of phase b can be calculated using the following formula:

[0033]

[0034] Where ρ n Let n represent the relative density value of material phase n in the finite element. In the above two-phase material optimization subproblem, material phases a and b are active phases, while other material phases are fixed phases and do not participate in the current optimization process.

[0035] Furthermore, in step 3, the computational model of the parameterized level set topology optimization method based on radial basis function interpolation is as follows:

[0036] Find:

[0037] Min:

[0038] St:

[0039] …

[0040]

[0041]

[0042]

[0043] Where N is the total number of material phases within a unit cell of the sandwich microstructure, and L is the number of control points on the horizontal set mesh within a unit cell of the sandwich microstructure. α represents the design variable at the microscale, indicating the expansion coefficient of the radial basis function constructed by the material phase n at the l-th node of the microscale horizontal set grid within the unit cell of the sandwich microstructure, and α represents the design variable of the unit cell of the sandwich microstructure. The vector of C; MI The objective function represents the overall flexibility of the core layer formed by the periodic arrangement of microstructure sandwich unit cells, u is the macroscopic displacement field, and ε is the objective function. ij and ε kl This represents the macroscopic strain field corresponding to the macroscopic displacement field. It is the macroscopic equivalent elastic tensor of the sandwich unit cell of a multi-material microstructure, i,j,k,l=1,2,…,d, where d represents the spatial dimension of the design problem; Ω s |Ω represents the structural domain contained in the middle core layer of a sandwich structure. s | indicates the volume of the structural domain contained in the intermediate core layer. This represents the volume constraint of phase n in the unit cell of the microstructure sandwich material. It is the relative density value of material phase n determined in step 2. It is a Heaviside function, a characteristic function used to characterize the structural form; It is the level set function corresponding to material phase n, which satisfies the following in the parameterized level set topology optimization method based on radial basis function interpolation: φ(x) is the radial basis function vector of the unit cell of the sandwich microstructure, α n (t) is the design variable for the unit cell of the sandwich microstructure. The vector; v represents the vector belonging to the kinematically permissible displacement space. The macroscopic virtual displacement field, f is the volume force of the macroscopic sandwich structure design domain Ω, and τ is the traction force acting on the boundary Γ; These are design variables that have been regularized to facilitate subsequent numerical implementation. and These are the preset design variables. The upper and lower boundaries.

[0044] Furthermore, The calculation is performed using a method based on homogenization theory, and the calculation model is as follows:

[0045]

[0046] in,

[0047]

[0048] , Let be the elastic tensor of the nth phase material, ij = 11, 22, 12, representing the horizontal, vertical, and shear directions, respectively. This refers to the strain field of the unit test in the pq direction. It refers to the unknown strain field caused by the unit test strain field in the pq direction, where Y represents the design domain of the microstructure sandwich unit cell, and |Y| represents the volume of the microstructure sandwich unit cell design domain. It is the elastic tensor of the nth phase material. It is the elastic tensor of the porous phase.

[0049] Furthermore, the alternating active phase algorithm is used to transform the multi-material layout optimization problem of the microstructure sandwich unit cell into multiple two-phase material layout optimization sub-problems for solution. The computational model of the two-phase material layout optimization sub-problems of the microstructure sandwich unit cell is as follows:

[0050] Find:

[0051] Min:

[0052] St:

[0053]

[0054]

[0055] in, a and b represent the a-th phase material and the b-th phase material in the two-phase material layout optimization subproblem, respectively. It is a design variable, representing the expansion coefficient of the radial basis function constructed by material phase a at the l-th node of the microscopic horizontal set grid within the design domain of the microstructure sandwich unit cell; It is the objective function of the sandwich unit cell of this microstructure. It is the volume constraint of material phase a in the microstructure sandwich unit cell. Va is the relative density value of material phase a in the microstructure sandwich unit cell, representing the volume fraction of material phase a in the microstructure sandwich unit cell, and V0 is the volume of the design domain of the microstructure sandwich unit cell. Ω is the level set function corresponding to material phase a, and Ω represents the overall design domain contained in the sandwich structure. ab This represents the design domain for material phases a and b in the microstructure sandwich unit cell; These are design variables after regularization. and They are presets The upper and lower boundary values; This represents the equivalent elastic tensor of the core microstructure containing material phase a and material phase b.

[0056] Compared with the prior art, the present invention, by adopting the above technical solution, has the following advantages:

[0057] 1. The technical solution provided by this invention, compared with existing technologies, achieves asymmetric design of sandwich structure geometry and materials. Specifically, on a macroscopic scale, it can not only obtain the optimal thickness of the upper and lower panels, but also the optimal material selection for the upper and lower panels; on a microscopic scale, it can obtain the optimal configuration of multi-material sandwich microstructure.

[0058] 2. This invention achieves optimized design of multi-material sandwich microstructures by using a topology optimization method based on parameterized level sets. The optimized multi-material sandwich microstructures have smooth structural boundaries and clear interfaces between material phases.

[0059] 3. This invention employs a multi-material topology optimization method based on the alternating active phase algorithm, which reduces the design variables and constraints for upper and lower panel optimization and sandwich microstructure optimization, effectively improving computational efficiency.

[0060] 4. This invention achieves joint optimization of the thickness of the upper and lower panels and the materials, as well as the topology of the sandwich microstructure, in a geometric-material asymmetric sandwich structure. Compared with traditional sandwich structure design, this invention greatly expands the design space of sandwich structure, fully utilizes the potential of materials, and can effectively improve the mechanical properties of sandwich structure and reduce structural weight.

[0061] The present invention will now be described in detail with reference to the accompanying drawings and embodiments. Attached Figure Description

[0062] Figure 1 This is a schematic diagram of a three-dimensional simply supported beam structure with a sandwich structure.

[0063] Figure 2 This is an initial design schematic diagram of a multi-material microstructure sandwich unit cell constructed according to a preferred embodiment of the present invention;

[0064] Figure 3 This is a schematic diagram showing the multi-material distribution of the upper panel, lower panel, and middle core layer of the sandwich structure obtained according to the optimized solution provided by the present invention.

[0065] Figure 4 This is a schematic diagram of the optimal topological configuration of a multi-material microstructure sandwich unit cell obtained according to the optimization scheme provided by the present invention;

[0066] Figure 5 This is a schematic diagram of a porous sandwich structure after the optimal configuration of the multi-material microstructure sandwich unit cell is arranged in a 3×3×5 periodic pattern.

[0067] Figure 6 This is a flowchart illustrating the implementation of the method of the present invention. Detailed Implementation

[0068] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0069] To achieve the above objectives, according to the present invention, a multi-scale topology optimization method for geometrically-material asymmetric sandwich structures is provided, the method comprising the following steps:

[0070] (a) For the sandwich structure to be optimized, it comprises a solid upper and lower panel and a core layer composed of periodically repeating porous core microstructures. The thickness and material type of the upper and lower panels, the material composition of the unit cell of the core layer microstructure, and its topological configuration are all determined by optimization calculation. First, the sandwich structure to be optimized is discretized into several slices, and the initial relative density of each material in each slice is set. The The value of is between (0,1); next, the alternating active phase algorithm is used to transform the N+1 phase multiphase material layout optimization problem into N(N+1) / 2 two-phase material layout optimization subproblems. Compared with the original N+1 phase multiphase material layout optimization problem, each two-phase material layout optimization subproblem contains fewer design variables and constraints, thus achieving higher optimization efficiency and easier algorithm implementation; subsequently, in each two-phase material layout optimization subproblem, the variable thickness method is used to optimize the density values ​​of each material in the slice layer; finally, based on the optimized material density of each slice layer... Based on the distribution results, a material density threshold f is set. s In this sandwich structure, the set of sliced ​​layers in which the density value of a certain material is greater than the threshold is regarded as the upper panel or the lower panel, so as to determine the thickness and material type of the upper and lower panels of the optimized sandwich structure. The set of sliced ​​layers in the sandwich structure in which the density value of each material is less than the threshold is regarded as the middle core layer of the sandwich structure, so as to determine the thickness of the middle core layer. The average density value of each material in each sliced ​​layer of the middle core layer is taken as the density value of each material in the core layer.

[0071] (b) Taking the intermediate core layer as the object to be optimized, the intermediate core layer is discretized into several periodic unit cells according to its thickness. The density value of each material in the core layer is used as the volume ratio constraint for the topological configuration optimization of the periodic unit cells. The multi-material parameterized level set topological optimization method is used to optimize the topology of the periodic unit cells. First, the topological distribution of the multi-material sandwich microstructure unit cells is described using a difference-based multi-material level set topological description model. Then, the equivalent mechanical properties of the multi-material sandwich microstructure unit cells are calculated using a homogenization method. Subsequently, since the core layer microstructure unit cells contain multiple materials, the alternating active phase algorithm is used to transform the multi-material layout optimization problem of the microstructure sandwich unit cells into multiple two-phase material layout optimization sub-problems. Finally, in each two-phase material layout optimization sub-problem, the multi-material parameterized level set topological optimization method is used to optimize the distribution of the two-phase material layout within the microstructure sandwich unit cells. By traversing all two-phase material layout optimization sub-problems, the optimal topological configuration of the multi-material microstructure sandwich unit cells is obtained.

[0072] (c) The multi-material microstructure sandwich unit cells obtained in step (b) are periodically repeated in the middle core layer of the sandwich structure to obtain the optimized complete structure of the middle core layer, thereby obtaining the optimized sandwich structure. Then, based on the equivalent mechanical properties of the multi-material microstructure sandwich unit cells calculated in step (b), the overall mechanical properties of the optimized sandwich structure are calculated. Based on this, it is determined whether the obtained sandwich structure meets the predetermined performance requirements. If not, return to step (b); if yes, the topology optimization process is implemented, and the optimized sandwich structure is output.

[0073] As a further preferred embodiment, in step (a), the calculation model for optimizing the relative density of each material in each slice layer of the sandwich structure using the variable thickness method is as follows:

[0074] Find:

[0075] Min:

[0076] St:G(ρ 1 )=V(ρ 1 -f1V0≤0

[0077] …

[0078] G(ρ n )=V(ρ n )-f n V0≤0

[0079]

[0080]

[0081]

[0082] Where N is the total number of material types used in the sandwich structure optimization, and M is the total number of sliced ​​layers after the sandwich structure is discretized. ε is a design variable at the macroscopic scale, representing the relative density value of material phase n contained in the m-th slice layer of the sandwich structure, wherein the relative density values ​​of all finite elements within a slice layer are consistent. C is the objective function, representing the structural flexibility value of the sandwich structure. Ω represents the total design domain encompassed by the sandwich structure, and ε ij and ε kl Let i, j, k, l = 1, 2, ..., d, where d is the spatial dimension. It is the elastic tensor of the finite element in the m-th layer of the sandwich structure, the Interpolation calculations are performed using the elastic tensor matrices of all materials within the m-th layer. G represents the volume fraction constraint of various materials in the sandwich structure, and f... n V is the volume fraction of a given material n. max This represents the maximum allowable volume fraction of the sandwich structure. u is the macroscopic displacement field, and v is the space within the allowable displacement area. The macroscopic virtual displacement field, f is the volume force acting on the design domain Ω of the sandwich structure, τ is the traction force acting on the boundary Γ, and ρ is the volume force acting on the boundary Γ. max =1 and ρ min =0.001 represent design variables The upper and lower boundaries of the value.

[0083] As a further preferred embodiment, in step (a), the The elastic tensor matrices of all materials within the m-th layer are calculated using the following interpolation model:

[0084]

[0085] Where E n It is the elastic tensor of the nth phase material.

[0086] As a further preferred embodiment, in step (a), since the multi-material layout optimization problem is a multi-constraint problem with many design variables and high optimization difficulty, the alternating active phase algorithm is used to transform the original N+1 phase multi-material layout optimization problem into N(N+1) / 2 two-phase material layout optimization subproblems. The original multi-phase material layout optimization problem includes N phase solid materials and 1 phase porous material. The optimization model of the two-phase material layout optimization subproblem is as follows:

[0087] Find:

[0088] Min:

[0089] St:G(ρ a )=V(ρ a )-f a V0≤0

[0090]

[0091]

[0092] 0≤ρ min ≤ρ a ≤ρ max

[0093] Where a and b represent material phase a and material phase b in the two-phase material layout optimization subproblem, respectively. It is a design variable at the macroscopic scale, representing the relative density value of material phase a in the m-th slice layer of the sandwich structure, C ab This is the objective function of the subproblem of optimizing the layout of the two-phase material. Based on the element elastic tensor... The interpolation model calculates the elastic tensor of a finite element containing material phases a and b. G(ρ a The volume constraint is for material phase a. In this two-phase material layout optimization subproblem, the design variable is the relative density ρ of material phase a. a The relative density of phase b can be calculated using the following formula:

[0094]

[0095] Where ρ n This represents the relative density value of material phase n in the finite element. In the above two-phase material optimization subproblem, material phases a and b are active phases, while other material phases are fixed phases and do not participate in the current optimization process.

[0096] As a further preferred embodiment, in step (b), the topological distribution of the core layer multi-material microstructure unit cell is described using the following difference-based multi-material level set topological description model:

[0097]

[0098] in, Let Ω represent the nth level set function. n express The physical area, Represents the solid region Ω n The boundary is defined by D, which is a fixed Eulerian reference space. x represents the coordinates of a point within space D, and t is the time variable describing the motion of the zero level plane of the level set function at any given moment. In the multi-material microstructure unit cell topological description model, the topological distribution of each material can be expressed through combinations of different level set functions. Therefore, using this multi-material microstructure unit cell topological description model, only N different level set functions are needed to describe the topological distribution of N+1 different phases of materials (including the porous phase).

[0099] As a further preferred embodiment, in step (b), according to the multi-material level set topological description model based on the difference set, the multi-material equivalent elastic tensor containing N+1 phase materials (including the porous phase) can be calculated using the following interpolation model:

[0100]

[0101] in, It is the elastic tensor of the nth phase material. It is the elastic tensor of the porous phase.

[0102] As a further preferred embodiment, in step (b), the macroscopic equivalent elastic tensor of the multi-material microstructure sandwich unit cell is... The calculation is performed using a method based on homogenization theory, and the calculation model is as follows:

[0103]

[0104]

[0105] in, Let ij = 11, 22, 12, representing the horizontal, vertical, and shear directions, respectively. kl, pq, and rs are similar to ij. This refers to the strain field of the unit test in the pq direction. It refers to the unknown strain field caused by the unit test strain field in the pq direction, where Y represents the design domain of the microstructure sandwich unit cell and Y represents the volume of the microstructure sandwich unit cell design domain.

[0106] As a further preferred embodiment, in step (b), the computational model of the core microstructure unit cell of the sandwich structure is optimized using a parameterized level set topology optimization method based on radial basis function interpolation as follows:

[0107] Find:

[0108] Min:

[0109] St:

[0110] …

[0111]

[0112]

[0113]

[0114] Where N is the total number of material phases within the unit cell of the sandwich microstructure, and L is the number of control points on the horizontal set mesh within the unit cell of the sandwich microstructure. It is a design variable at the microscale, representing the expansion coefficient of the radial basis function constructed by the material phase n within the unit cell of the sandwich microstructure at the l-th node of the micro-level set grid, and α represents the design variable of the unit cell of the sandwich microstructure. The vector of C; MI The objective function represents the overall flexibility of the core layer formed by the periodic arrangement of the microstructure sandwich unit cells, u is the macroscopic displacement field, and ε is the objective function. ij and ε kl This represents the macroscopic strain field corresponding to the macroscopic displacement field. It is the macroscopic equivalent elastic tensor of the sandwich unit cell of a multi-material microstructure, i,j,k,l=1,2,…,d, where d represents the spatial dimension of the design problem; Ω s |Ω represents the structural domain contained in the middle core layer of the sandwich structure. s | indicates the volume of the structural domain contained in the intermediate core layer. This indicates the volume constraint of phase n of the microstructure sandwich unit cell material. It is the relative density value of material phase n in the core layer microstructure sandwich unit cell determined in step (a). It is a Heaviside function, a characteristic function used to characterize the structural form; It is the level set function corresponding to material phase n, which satisfies the following in the parameterized level set topology optimization method based on radial basis function interpolation: φ(x) is the radial basis function vector of the unit cell of the sandwich microstructure, α n (t) is the unit cell design variable of the sandwich microstructure. The vector; v represents the vector belonging to the kinematically permissible displacement space. The macroscopic virtual displacement field, f is the volume force of the macroscopic sandwich structure design domain Ω, and τ is the traction force acting on the boundary Γ; These are design variables that have been regularized to facilitate subsequent numerical implementation. and These are design variables The upper and lower boundaries.

[0115] As a further preferred embodiment, in step (b), the alternating active phase algorithm is also used to transform the multi-material layout optimization problem of the core layer microstructure sandwich unit cell into multiple two-phase material layout optimization sub-problems for solution. The computational model of the two-phase material layout optimization sub-problems of the core layer microstructure sandwich unit cell is as follows:

[0116] Find:

[0117] Min:

[0118] St:

[0119]

[0120]

[0121] Where a and b represent material phase a and material phase b in the two-phase material optimization subproblem of the core microstructure, respectively. It is a design variable, representing the expansion coefficient of the radial basis function constructed at the l-th node of the microscopic horizontal set grid within the design domain of the core layer microstructure sandwich unit cell; It is the objective function of the core layer microstructure sandwich unit cell. It is the volume constraint of material phase a in the core unit cell of the core layer microstructure sandwich structure. Va is the relative density value of material phase a in the core layer microstructure sandwich unit cell, representing the volume fraction of material phase a in the core layer microstructure sandwich unit cell, and V0 is the volume of the design domain of the core layer microstructure sandwich unit cell. Ω is the level set function corresponding to material phase a, and Ω represents the overall design domain contained in the sandwich structure. ab This represents the design domain of material phases a and b in the core unit cell of the core microstructure. These are design variables after regularization. and They are The upper and lower boundaries; This represents the equivalent elastic tensor of the core microstructure containing material phase a and material phase b. Based on the macroscopic equivalent elastic tensor calculation model of the multi-material microstructure sandwich unit cell, it can be derived from... and Interpolation calculation yielded:

[0122]

[0123] in

[0124] As a further preferred embodiment, in step (a), a gradient-based optimization algorithm is used to update the design variables. Therefore, it is necessary to calculate the objective function and constraint function for the design variables. The sensitivity information is calculated using the following formula:

[0125] Sensitivity of the objective function to design variables:

[0126]

[0127] Sensitivity of constraint functions to design variables:

[0128] Where ζ represents the sensitivity of the overall flexibility of the sandwich structure to all design variables within the design domain, mean(ζ,P) represents the average value of ζ along the direction parallel to the slice layer, and P represents the direction parallel to the slice layer. Based on the sensitivity information, it can be seen that all macroscopic units within each optimized slice layer have the same material density. Furthermore, the material density of all slice layers exhibits a gradient distribution in the direction perpendicular to the slice layer.

[0129] As a further preferred embodiment, in step (b), a gradient-based optimization algorithm is used to update the design variables, and the sensitivity information is calculated using the following formula:

[0130] Sensitivity of the objective function to design variables:

[0131]

[0132]

[0133] in It is the differential of the Heaviside function, and φ(x) is the compactly supported radial basis function.

[0134] Sensitivity of volume constraints to design variables:

[0135] Figure 1 This is a schematic diagram of the design domain, loads, and boundary conditions of the simply supported beam sandwich structure to be optimized in this embodiment. For lack of generality, all physical quantities used in this embodiment are assumed to be dimensionless. The design domain of the sandwich structure has dimensions of length L = 40, width W = 10, and height H = 10. A concentrated load F = 100 is applied to the center of the upper surface of the sandwich structure. The sandwich structure is meshed using 40 × 10 × 10 = 4000 hexahedral finite elements. The sandwich structure is designed using two materials with elastic moduli of E1 = 10 (referred to as the weak material) and E2 = 100 (referred to as the strong material), respectively, and Poisson's ratio of ν = 0.3 for both. The usage of the two materials in the sandwich structure design is V1 = 50% and V2 = 30%, respectively.

[0136] Figure 2 This is the initial design of the multi-material microstructure sandwich unit cell to be optimized in this embodiment, which is composed of the two materials mentioned above.

[0137] Figure 3 This is a schematic diagram of the multi-material distribution results of the upper panel, lower panel, and middle core layer of the sandwich structure optimized using the variable thickness method. It can be seen that the thickness of the upper panel of the optimized sandwich structure is 1, which is composed of only strong material; the thickness of the lower panel is 2, which is composed of one layer of strong material and one layer of weak material. This choice of thickness and material type of the upper and lower panels is very meaningful for the three-dimensional simply supported beam structure, because in the three-dimensional simply supported beam sandwich structure, the lower panel has the dominant advantage in resisting bending deformation.

[0138] Figure 4 The optimal topological configuration of the multi-material microstructure sandwich unit cell is obtained by using a parameterized level set topological optimization method based on tight-support radial basis function interpolation. The optimal configuration of the multi-material microstructure sandwich unit cell contains 57.14% weak material and 14.29% strong material, and is determined by calculation based on the optimization results of the aforementioned upper and lower panels.

[0139] Figure 5 The porous sandwich structure is formed by arranging the optimal configuration of the multi-material microstructure sandwich unit cell in a 3×3×5 periodic pattern. It can be seen that the porous microstructure sandwich unit cells have good connectivity.

[0140] Figure 6 This is a flowchart illustrating the implementation of a multi-scale topology optimization method for a geometry-material asymmetric sandwich structure according to a preferred embodiment of the present invention.

[0141] The above description provides examples of the preferred embodiments of the present invention. Parts not detailed herein are common knowledge to those skilled in the art. The scope of protection of the present invention is determined by the claims. Any equivalent modifications based on the technical teachings of the present invention are also within the scope of protection of the present invention.

Claims

1. A multi-scale topology optimization design method for geometrically-material asymmetric sandwich structures, characterized in that, Includes the following steps: Step 1: The sandwich structure to be optimized consists of a solid panel and a core layer composed of a periodically repeating porous core microstructure. The sandwich structure to be optimized is first discretized into several slices of equal thickness, and the relative density of each material in each slice is set. initial value, The value of is between (0,1), m is the sequence number of the sandwich layer, n is the sequence number of the material type, n=1,2,…,N,m=1,2,…,M,N is the total number of material types used in the sandwich structure optimization, and M is the total number of slices after the sandwich structure is discretized. Step 2: Optimize the model using the variable thickness method to obtain the relative density of each material in each slice of the sandwich structure. The calculation results of the relative density of each slice have two possibilities:

1. The relative density of a certain material is 1, and the relative densities of other materials are all 0; 2. The relative densities of all materials are greater than or equal to 0 and less than 1. In step 2, the specific calculation model of the variable thickness optimization method is as follows: St:G(r 1 )=V(ρ 1 )-f1V0≤0 … G(r n )=V(ρ n )-f n V0≤0 in, Let represent the relative density value of the nth phase material contained in the m-th slice layer of the sandwich structure. The relative density values ​​of all finite elements within a slice layer are kept consistent. C is the objective function, represents the structural compliance value of the sandwich structure, Ω represents the total design domain contained in the sandwich structure, and ε ij and ε kl Let i, j, k, l = 1, 2, ..., d, where d is the spatial dimension. It is the elastic tensor of the finite element in the m-th layer of the sandwich structure. Interpolation calculations are performed using the elastic tensor matrices of all materials within the m-th layer, where G represents the volume fraction constraint of various materials in the sandwich structure, and f... n V is the volume fraction of the given nth phase material. max This represents the maximum allowable volume fraction of the sandwich structure, u is the macroscopic displacement field, and v is the space within the allowable displacement. The macroscopic virtual displacement field, f is the volume force acting on the design domain Ω of the sandwich structure, τ is the traction force acting on the boundary Γ, and ρ is the volume force acting on the boundary Γ. max and ρ min These are the upper and lower boundaries of the preset design variable values. If the obtained values ​​are... The lower limit value ρ min Then it is determined =0; The sliced ​​layer in case 1 is used as the solid surface layer, and a material with a relative density of 1 is directly selected as the material for the corresponding solid surface layer. The sliced ​​layer in case 2 is used as the core layer. The arrangement of the solid surface layers and core layers is obtained. The relative density value of each material is obtained by summing the relative densities of each material in all core layers and averaging the results. n is the material type number; Step 3: Take the core layer as the object to be optimized, and discretize the core layer into several periodic microstructure sandwich units according to the thickness of the core layer. Use the relative density value of each material in the core layer as the volume ratio constraint for the topology optimization of the periodic microstructure sandwich unit. Use the parameterized level set topology optimization method based on radial basis function interpolation to perform topology optimization on the microstructure sandwich unit to obtain the optimal distribution of each material in the design domain of the microstructure sandwich unit. The combination of the optimal distribution of each material in the design domain of the microstructure sandwich unit is the optimal topology of the microstructure sandwich unit. In step 3, the computational model of the parameterized level set topology optimization method based on radial basis function interpolation is as follows: … Where N is the total number of material phases within a unit cell of the sandwich microstructure, and L is the number of control points on the horizontal set mesh within a unit cell of the sandwich microstructure. α represents the design variable at the microscale, indicating the expansion coefficient of the radial basis function constructed by the material phase n at the l-th node of the microscale horizontal set grid within the unit cell of the sandwich microstructure, and α represents the design variable of the unit cell of the sandwich microstructure. The vector of C; MI The objective function represents the overall flexibility of the core layer formed by the periodic arrangement of microstructure sandwich unit cells, u is the macroscopic displacement field, and ε is the objective function. ij and ε kl This represents the macroscopic strain field corresponding to the macroscopic displacement field. It is the macroscopic equivalent elastic tensor of the sandwich unit cell of a multi-material microstructure, i,j,k,l=1,2,…,d, where d represents the spatial dimension of the design problem; Ω s |Ω represents the structural domain contained in the middle core layer of a sandwich structure. s | indicates the volume of the structural domain contained in the intermediate core layer. This represents the volume constraint of phase n in the unit cell of the microstructure sandwich material. It is the relative density value of material phase n determined in step 2. It is a Heaviside function, a characteristic function used to characterize the structural form; It is the level set function corresponding to material phase n, which satisfies the following in the parameterized level set topology optimization method based on radial basis function interpolation: φ(x) is the radial basis function vector of the unit cell of the sandwich microstructure, α n (t) is the design variable for the unit cell of the sandwich microstructure. The vector; v represents the vector belonging to the kinematically permissible displacement space. The macroscopic virtual displacement field, f is the volume force of the macroscopic sandwich structure design domain Ω, and τ is the traction force acting on the boundary Γ; These are design variables that have been regularized to facilitate subsequent numerical implementation. and These are the preset design variables. The upper and lower bounds; Step 4: Periodically repeat the optimal topological configuration of the microstructure sandwich unit cell obtained in the previous step in the middle core layer of the sandwich structure to obtain the optimized complete structure of the middle core layer. Arrange the solid surface layer and core layer according to the arrangement method obtained in Step 1, and determine the solid surface layer material according to the preset requirements to obtain the optimized complete sandwich structure.

2. The multi-scale topology optimization design method for geometric-material asymmetric sandwich structures according to claim 1, characterized in that, The The elastic tensor matrices of all materials within the m-th layer are calculated using the following interpolation model: Where E n It is the elastic tensor of the nth phase material.

3. The multi-scale topology optimization design method for geometric-material asymmetric sandwich structures according to claim 1, characterized in that, In step 2, the alternating active phase algorithm is used to transform the original N+1 phase multi-material layout optimization problem into N(N+1) / 2 two-phase material layout optimization sub-problems. The original multi-phase material layout optimization problem includes N phase solid materials and 1 phase porous material. The optimization model for the two-phase material layout optimization sub-problems is as follows: St:G(r a )=V(ρ a )-f a V0≤0 0≤ρ min ≤ρ a ≤ρ max Where a and b represent the a-th phase material and the b-th phase material in the two-phase material layout optimization subproblem, respectively, C ab It is the objective function of the two-phase material layout optimization subproblem, based on the element elasticity tensor. The interpolation model calculates the elastic tensor of a finite element containing material phases a and b. G(ρ a The volume constraint of material phase a is given by ( ). In this two-phase material layout optimization subproblem, the design variable is the relative density ρ of material phase a. a The relative density of phase b can be calculated using the following formula: Where ρ n Let n represent the relative density value of material phase n in the finite element. In the above two-phase material optimization subproblem, material phases a and b are active phases, while other material phases are fixed phases and do not participate in the current optimization process.

4. The multi-scale topology optimization design method for geometric-material asymmetric sandwich structures according to claim 1, characterized in that, The calculation is performed using a method based on homogenization theory, and the calculation model is as follows: in, Let be the elastic tensor of the nth phase material, ij = 11, 22, 12, representing the horizontal, vertical, and shear directions, respectively. This refers to the strain field of the unit test in the pq direction. It refers to the unknown strain field caused by the unit test strain field in the pq direction, where Y represents the design domain of the microstructure sandwich unit cell, and |Y| represents the volume of the microstructure sandwich unit cell design domain. It is the elastic tensor of the nth phase material. It is the elastic tensor of the porous phase.

5. The multi-scale topology optimization design method for geometrically-material asymmetric sandwich structures according to claim 4, characterized in that, The alternating active phase algorithm is used to transform the multi-material layout optimization problem of microstructure sandwich unit cells into multiple two-phase material layout optimization sub-problems for solution. The computational model of the two-phase material layout optimization sub-problems of microstructure sandwich unit cells is as follows: in, a and b represent the a-th phase material and the b-th phase material in the two-phase material layout optimization subproblem, respectively. It is a design variable, representing the expansion coefficient of the radial basis function constructed by material phase a at the l-th node of the microscopic horizontal set grid within the design domain of the microstructure sandwich unit cell; It is the objective function of the sandwich unit cell of this microstructure. It is the volume constraint of material phase a in the microstructure sandwich unit cell. Va is the relative density value of material phase a in the microstructure sandwich unit cell, representing the volume fraction of material phase a in the microstructure sandwich unit cell, and V0 is the volume of the design domain of the microstructure sandwich unit cell. Ω is the level set function corresponding to material phase a, and Ω represents the overall design domain contained in the sandwich structure. ab This represents the design domain for material phases a and b in the microstructure sandwich unit cell; These are design variables after regularization. and They are presets The upper and lower boundary values; This represents the equivalent elastic tensor of the core microstructure containing material phase a and material phase b.

Images