CN111027150A - Multi-component topology optimization design and processing method and system for microstructure product - Google Patents

Abstract

The invention discloses a multi-component topology optimization design and processing method and a system for a microstructure product, which are used for constructing a pseudo-density design variable phi and a multi-component design vector (mu)¹,μ²,….,μ^k,….,μ^K) Taking the minimum compliance as an objective function, and considering material use volume ratio constraint, size constraint of each component, local average density constraint, multi-component solid interface and connecting part volume constraint; solving the objective function to obtain phi and mu^kThe optimal value of (2); according to phi, mu^kDetermining whether the material is arranged at each position in the design domain omega, and forming a microstructure product model by all parts of the material arranged in the design domain omega. In the design process of the microstructure product, the size constraint of the processing method, the assembly connection mode of the components and the physical properties of the connection part of the components in the processing process of the microstructure product are considered, the method is a design and manufacture integrated method, and can meet the high-precision processing requirement of large components of complex products.

Description

A multi-component topology optimization design, processing method and system for a microstructure product

technical field

The invention belongs to the fields of topology optimization, microstructure product design and processing, functional gradient microstructure, design and manufacturing integration and 3D printing, and particularly relates to a multi-component topology optimization design, processing method and system for a microstructure product.

Background technique

The microstructure not only has the characteristics of light weight and excellent performance, but also can realize many functions such as negative Poisson's ratio, zero thermal expansion, high heat dissipation ratio and performance ratio, as well as frequency band sound absorption performance and light guidance, so it is widely used in automobiles, ships, trains and so on. There is a great demand for industrial applications in the fields of aerospace and aerospace.

The existing design methods of microstructure products (products composed of microstructures, including large components composed of microstructures) are mainly designed by designing microstructure elements, and then by adding periodic boundary conditions, using finite element simulation analysis, and finally flattening the microstructure elements. Paving way to achieve microstructured products. This type of method is a tentative design method, which requires repeated attempts to design different microstructure units, and repeated experiments to obtain microstructure units with better performance and microstructure products that meet performance requirements, which consumes a lot of manpower and material resources. In recent years, various methods of topology optimization have been applied to the field of microstructure product design. However, the existing microstructure product design method based on topology optimization has problems such as the connection of microstructure units is not smooth, and requires a lot of post-processing to ensure the smoothness of the connection between microstructure units. However, the post-processing process will inevitably lose the physical properties of the optimized structure. . Moreover, due to the different shapes of microstructure products designed by topology optimization, traditional processing methods basically cannot realize the processing of microstructure products designed by topology optimization or the processing cost is huge, so the existing microstructure products are mostly processed by 3D printing. However, existing 3D printers have size limitations, especially for metal materials, there are still problems of deformation and fracture due to excessive residual stress during printing of large components (large-sized components). Therefore, processing large-sized microstructure products has become a current research difficulty. It is also the key that restricts the application of microstructure in large components of complex products in the fields of automobiles, ships, trains and aerospace.

The multi-component topology optimization method in the large component structure design method is to optimize the large component into multiple small components that meet the size constraints of the processing method (such as the maximum processing size of a 3D printer), and then assemble multiple small components to obtain the large component. However, the existing multi-component topology optimization methods are mainly aimed at the optimization of solid components and are directly applied to large components composed of microstructures. There are great challenges in connecting parts of microstructural units of multiple building blocks. At the same time, the current multi-component topology optimization methods based on the gradient method do not consider the physical properties of the component connection part and the manufacturing process constraints, while the traditional multi-component topology optimization method using the genetic algorithm considers the physical properties of the component connection part, but the computational efficiency Very low, unable to meet the optimization requirements of thousands or even hundreds of thousands of design variables.

Therefore, in order to meet the needs of the industrial application of microstructures in large components, there is an urgent need for a topology optimization design method for microstructure products that integrates design and manufacture.

SUMMARY OF THE INVENTION

The technical problem solved by the present invention is to provide a multi-component topology optimization design, processing method and system for microstructure products in view of the deficiencies of the prior art, which can realize the topology optimization design and processing of microstructure products with high precision.

A multi-component topology optimization design method for a microstructure product, comprising the following steps:

Step 1. Set the design domain of topology optimization as Ω; construct a pseudo-density design variable (topological geometry design variable) φ, φ is a continuous function in the design domain Ω, and the value of φ at each position in the design domain Ω represents the design respectively. Whether the material is arranged at the corresponding position in the domain Ω, -1≤φ≤1; construct a multi-component design vector (μ ¹ , μ ² , ...., μ ^k , ...., μ ^K ), where μ ^k represents The kth component design variable, μ ^k is a continuous function in the design domain Ω, the value of μ ^k at each position in the design domain Ω represents the possibility that the corresponding position in the design domain Ω belongs to the kth component, 0≤ μ ^k ≤ 1, k=1, 2, ...., K, K is the maximum number of components that divide the design domain into;

Step 2. Build the following objective function:

Among them, c(φ) is the compliance function, S is the total stiffness matrix of the structure (each element in the design domain Ω has a stiffness matrix, which is organized together to form the overall stiffness matrix of the structure); U is the node displacement vector, whose dimension is equal to node, the element of each dimension corresponds to the displacement of a node; F is the node equivalent load vector, whose dimension is equal to the number of nodes, and the element of each dimension corresponds to the equivalent load of a node;

V is the volume fraction of the material used; V ⁰ is the upper limit of the volume fraction of the material (maximum constraint);

为第k个组件的最大球形包围盒半径；R_max为各组件尺寸上限值(最大值约束)，根据加工方法尺寸约束进行设定；

is the maximum spherical bounding box radius of the kth component; R _max is the upper limit of the size of each component (maximum constraint), which is set according to the size constraint of the processing method;

P _l is the p-norm approximation of the average pseudo-density design variable ρ _l , P _max is the upper limit of the local average density (maximum constraint), 0≤P _max ≤1;

C is the volume of the multi-component entity interface and the connected part, and C ⁰ is the upper limit of the volume of the multi-component entity interface and the connected part (maximum constraint);

Each upper limit is artificially set according to needs/experience;

Step 3, solve the objective function; set the component assembly connection mode and the physical properties of the component connection part, for each group of φ, μ ^k values, adopt the finite element analysis method (discrete the design domain Ω into multiple elements, the vertices of each element is the node) to obtain the corresponding U, S, F, and the corresponding objective function value; iteratively solve to obtain the optimal value of φ, μ ^k ;

Step 4. According to the optimal values of φ and ^μk , determine whether materials are arranged at each position in the design domain Ω, and a group microstructure product model is composed of all parts of the material arranged in the design domain Ω.

In the design process of the microstructure product, the above method takes into account the size constraints of the processing method in the process (manufacturing) of the microstructure product, and is an integrated method of design and manufacture, which can meet the high-precision processing requirements of large components of complex products.

Further, the material uses a volume fraction V= _∫Ω ∑ _k ρm ^k dΩ, that is, ∑ _k ρm ^k is integrated in the entire design domain;

Among them, ρ is a pseudo-density design variable with basically only two values of 0 and 1 in the design domain, which is obtained by the following steps:

First, the pseudo-density design variable φ is smoothed using the Helmhertz partial differential equation, as follows:

为梯度算子，

为光顺后的伪密度设计变量；r_ρ为光顺半径，同时能够控制拓扑优化结构最小几何尺寸，r_ρ为经验参数，一般取值为大于加工方式的最小加工精度；in the formula

is the gradient operator,

is the pseudo-density design variable after smoothing; r _ρ is the smoothing radius, which can control the minimum geometric size of the topology optimization structure, and r _ρ is an empirical parameter, which is generally greater than the minimum machining accuracy of the machining method;

获得设计域Ω内基本上只有0和1两种取值的伪密度设计变量ρ，公式如下：Then, using the step projection function

The pseudo-density design variable ρ, which basically has only two values of 0 and 1 in the design domain Ω, is obtained. The formula is as follows:

In the formula, h is the control parameter of the continuous transition part of the step projection, which is an empirical parameter, and the general value is 0.5;

m ^k is the k-th component variable of each unit in the design domain, m ^k ∈ {0, 1}, obtained through the following steps:

其取值范围为

公式如下：First, the variable μ ^k is smoothed by using the Helmhertz partial differential equation to obtain the smoothed variable

Its value range is

The formula is as follows:

Then, the DMO projection method is used to obtain m ^k , and the specific processing formula is as follows:

Among them, r _m is the multi-component interface width control parameter, which is an empirical parameter, and its value is set according to the width of the component connection interface required by the design; P _m is the penalty coefficient, which is an empirical parameter, generally between 6-15;

The above steps can make the variable m ^k obtained after projection basically take the value of 0 or 1; when a certain position in the design domain Ω belongs to the kth component, its corresponding m ^k =1, and its corresponding m ⁱ =0, i ∈ {1, 2, ...., K}, and i≠k.

采用R_a近似，

式中r为半径，即第k个组件上各节点到该组件中心的距离，r_c为第k个组件的中心坐标，可以表示为：Further, the

Using the _Ra approximation,

where _r is the radius, that is, the distance from each node on the kth component to the center of the component, and rc is the center coordinate of the kth component, which can be expressed as:

p is an empirical parameter, generally a number between 6-10.

Further, described P ₁ is solved by the following steps:

First, the pseudo-density design variable ρ is processed by the variable-radius Helmhertz partial differential equation to obtain the average pseudo-density design variable ρ _l in the local area. The formula is as follows:

Among them, r _l is the average density neighborhood control variable, and its value is greater than r _ρ ;

Then, calculate the average pseudo-density design variable _ρl based on the p-norm approximation:

where P _l is the p-norm approximation of the mean pseudo-density design variable p _l .

Further, the multi-component solid interface and the connected part volume C are calculated by the following formula:

In the formula, g(ρ _l ) is a linear fitting or spline interpolation function; M ^k is the physical interface of the kth component, obtained through the following steps:

其取值范围为

公式如下：First, the variable m ^k is smoothed by using the Helmhertz partial differential equation to obtain the smoothed variable

Its value range is

The formula is as follows:

In the formula, _rs ^k is the interface width control parameter of the kth component, which is an empirical parameter, and its value is set according to the component connection interface width required by the design;

Then, the step projection function is used to obtain the parameter ω ^k with a value of 0 or 1 in the design domain:

值；In the formula, tanh( ) represents the hyperbolic tangent function; β and η are the control parameters of the step projection function, and β generally takes 8-16, which is mainly used to control the steepness of the transition part of the step projection function 0-1, and η is used as When the control step projection function value is equal to 0.5, the corresponding

value;

Then, the entity interface M ^k of the kth component can be constructed by a simple product method:

M ^k =(1-m ^k )w ^k ;

I _ij is the physical property of the i-th and j-th components from the entity component part to the connecting part between them, and its mathematical expression is:

Among them, D ₁ and D ₂ are the physical properties of the entity component part and the physical properties of the connection part, respectively. I _ij is the physical model of multi-component assembly connection.

The present invention also provides a multi-component processing method for a microstructure product, comprising the following steps:

Step i, using the above-mentioned multi-component topology optimization microstructure design method, to solve the optimal value of φ, μ ^k , and its corresponding pseudo-density design variable ρ, variable m ^k , average pseudo-density design variable ρ _l and parameters ω ^k ;

Step ii, obtain the mathematical model of each component constituting the microstructure according to the following formula:

C ^k =ρm ^k +g(ρ _l )(1-m ^k )ω ^k

Among them, C ^k represents the kth component that constitutes the microstructure;

Step iii, using processing equipment to process all components according to the mathematical model of each component of the microstructure;

Step iv, assembling each component to obtain a microstructure product.

The present invention also provides a multi-component topology optimization design system for microstructure products, including a memory and a processor, wherein the memory stores a computer program, characterized in that, when the computer program is executed by the processor, all The processor implements any of the above-mentioned multi-component topology optimization design methods for microstructure products.

The present invention also provides a computer-readable storage medium on which a computer program is stored, characterized in that, when the computer program is executed by a processor, any one of the above-mentioned methods for optimizing multi-component topology of a microstructure product is implemented.

The present invention also provides a multi-component processing system for microstructure products, including a memory and a processor, wherein the memory stores a computer program, and is characterized in that it also includes a processing device;

When the computer program is executed by the processor, the processor enables the processor to implement steps i to ii in the above method;

The processing equipment processes all the components according to the mathematical model of each component of the microstructure obtained by the processor;

The individual components are assembled to obtain a microstructured product.

The invention discloses a multi-component topology optimization design, processing method and system for a microstructure product, which mainly include functional gradient microstructure optimization design (that is, optimizing pseudo-density design variables), multi-component optimal segmentation (that is, optimizing multi-component design vectors) , multi-component connection interface physical model construction and optimization, etc. The model can be divided into three layers. The first layer is the basic design parameters, including pseudo-density design variables (density variables) and component design variables (component variables); the second layer is intermediate variables, including multi-component entity interface and multi-component Assembly connection physical modeling, etc.; third-layer structure design output, including machinable multi-components and final assembly results, the model frame is shown in Figure 1, and K=3 is used as an example in the figure. Through topology optimization, the optimization result obtained is the microstructure product in the design domain, such as the black part of the geometric structure shown in Figure 1, the black part in the figure is the part where the material is arranged, and the white part is empty and no material is arranged.

Beneficial effects:

The invention provides a multi-component topology optimization design, processing method and system for microstructure products, and a computer-readable storage medium, which can realize the topology optimization design of microstructure products with high precision; and has the following advantages:

(1) The microstructure in the microstructure product is designed by the topology optimization method. Compared with the existing microstructure tentative design method, the human and material resources are less expensive;

(2) The smooth connection between microstructure units and components (between microstructure units and between microunits and components) is realized by the variable radius Helmhertz partial differential equation smoothing method combined with variable density topology optimization. Topological optimization design; the microstructure product designed by the present invention is composed of a functionally graded microstructure (Functionally graded lattice structure), which comprises a plurality of microstructure units with similar sizes and shapes, and the size and shape of the microunits at different positions are slightly different. Meet the performance requirements of the corresponding location;

(3) Using the cladding modeling method (two-step smoothing and projection method), the multi-component entity interface of the microstructure product is constructed, and the physical model of the multi-component assembly connection is further constructed, which can ensure the multi-scale between the microstructure unit and the components. Smooth connection, and at the same time, simulation of component assembly methods including gluing, welding and riveting is realized through assembly connection modeling to realize the integration of microstructure product design and manufacturing;

(4) The post-processing stage in the traditional microstructure topology optimization design is not required, and there will be no physical performance loss in the post-processing stage;

(5) In the process of microstructure product design, the size constraints of the processing method in the process of microstructure product processing (manufacturing) are considered, and the multi-component topology optimization design method is adopted, which can realize the integration of design and manufacture, and can meet the size constraints of additive manufacturing equipment , which can meet the high-precision processing requirements of large components of complex products; it does not need to rely on large-scale additive manufacturing equipment for processing, avoids a large amount of processing costs, and greatly reduces the risk of defects and residual stress deformation during the additive manufacturing process of large components. There are small additive equipment or traditional processing methods to build large components of complex products with high precision and high quality. The invention has important practical significance for the specific application of the microstructure in the fields of automobiles, ships, airplanes, high-speed trains and the like.

(6) There is no fixed element type, the microstructure is automatically generated in the topology optimization process based on the pseudo-density design variables, and will not be constrained by the limited search space caused by the fixed initial shape of the microstructure element in the traditional design and optimization method, and the optimized microstructure product The minimum flexibility of , is very close to the results of topology optimization of solid structures by the variable density method.

Description of drawings

Figure 1 is the framework of the topology optimization model for microstructure products (3 components are taken as an example);

Figure 2 is a schematic diagram of a two-step smoothing and projection method to build a component entity interface;

Figure 3 shows the geometry and physical modeling of connections between components;

Figure 4 shows the cantilever beam topology optimization design domain and boundary conditions

Fig. 5 is the topology optimization of the microstructure product of the assembly of weakly connected 2 components, in which Fig. 5(a) is the component variable 1, Fig. 5(b) is the component variable 2, and Fig. 5(c) is the optimized machinable component 1. Fig. 5(d) is the optimized machinable component 2, Fig. 5(e) is the optimized assembly result of the microstructure, and Fig. 5(f) is the multi-component entity interface and the assembly connection part (the gray part is the material, the black part is empty)

Fig. 6 is the topology optimization of the microstructure product of the assembly strong connection 2 component, in which Fig. 6(a) is the component variable 1, Fig. 6(b) is the component variable 2, and Fig. 6(c) is the optimized machinable component 1. Fig. 6(d) is the optimized machinable component 2, and Fig. 6(e) is the optimized assembly result of the microstructure, of which Fig. 6(f) is the multi-component entity interface and the assembly connection part (the gray part is the material, the black part is empty). )

Fig. 7 shows the topology optimization of the assembly microstructure product with medium connection strength and small machining size constraint 3 components, in which Fig. 7(a) is the component variable 1, Fig. 7(b) is the optimized machinable component 1, and Fig. 7(c) is the Component variant 2, Fig. 7(d) is the optimized machinable component 2, Fig. 7(e) is the component variant 3, Fig. 7(f) is the optimized machinable component 3, and Fig. 7(g) is the microstructure Optimized assembly results, Figure 7(h) is the multi-component entity interface and assembly connection part

Figure 8 shows the topology optimization of the microstructure product of the assembly with medium connection strength, large processing size constraints Component variant 2, Fig. 8(d) is the optimized machinable component 2, Fig. 8(e) is the component variant 3, Fig. 8(f) is the optimized machinable component 3, and Fig. 8(g) is the microstructure The optimized assembly results, Figure 8(h) is the multi-component entity interface and assembly connection part.

Detailed ways

The present invention will be described in detail below with reference to the accompanying drawings and specific examples.

Example 1:

The present embodiment discloses a multi-component topology optimization design method for a microstructure product, which includes the following steps:

Step 1: Set the design domain of topology optimization as Ω; construct a pseudo-density design variable (topological geometric design variable) φ, φ is a continuous function in the design domain Ω, and the value of φ at each position in the design domain Ω represents the design respectively. Whether the material is arranged at the corresponding position in the domain Ω, -1≤φ≤1; construct a multi-component design vector (μ ¹ , μ ² , ...., μ ^k , ...., μ ^K ), where μ ^k represents The kth component design variable, μ ^k is a continuous function in the design domain Ω, the value of μ ^k at each position in the design domain Ω represents the possibility that the corresponding position in the design domain Ω belongs to the kth component, 0≤ μ ^k ≤ 1, k=1, 2, ...., K, K is the maximum number of components to divide the design domain into (manually set); the design domain obtained by the final optimization is the multi-component corresponding to each position in Ω In the design vector, only one element is 1, and the other elements are 0; if in the multi-component design vector corresponding to a certain position obtained by the final optimization, μ ^k =1, indicating that this position belongs to the kth component;

Step 2: Use the Helmhertz partial differential equation to smooth the pseudo-density design vector φ, so as to solve the checkerboard problem existing in the optimization process. The formula is as follows:

为梯度算子，

为光顺后的伪密度设计向量；r_ρ为光顺半径，同时能够控制拓扑优化结构最小几何尺寸，r_ρ为经验参数，一般取值为大于加工方式的最小加工精度；in,

is the gradient operator,

is the pseudo-density design vector after smoothing; r _ρ is the smoothing radius, which can control the minimum geometric size of the topology optimization structure, r _ρ is an empirical parameter, and the value is generally greater than the minimum machining accuracy of the machining method;

为[-1，1]范围的变量，然后采用阶跃投影函数

获得设计域内基本上只有0和1两种取值的伪密度设计变量ρ，ρ包括0和1两种取值(0代表没有材料，1代表有材料)，以及两者中间一个很窄的过度带，以实现光顺连接，得以用梯度法优化，便于加工制造，具体公式如下：Step 3. Pseudo-density design vector after smoothing

is a variable in the range [-1, 1] and then uses a step projection function

Obtain the pseudo-density design variable ρ, which basically has only two values of 0 and 1 in the design domain. ρ includes two values of 0 and 1 (0 means no material, 1 means material), and a narrow transition between the two To achieve smooth connection, it can be optimized by gradient method, which is convenient for processing and manufacturing. The specific formula is as follows:

Among them, h is the control parameter of the continuous transition part of the step projection, which is an empirical parameter, and the general value is 0.5.

Step 4: Process the pseudo-density design vector ρ through the variable-radius Helmhertz partial differential equation to obtain the average pseudo-density design vector ρ _l in the local area, as follows:

Among them, r _l is the average density neighborhood control variable, which is an empirical parameter, and its value is greater than r _ρ .

Step 5. By adding the approximate maximum value constraint P _max based on the p-norm (p is an empirical parameter, generally taking a number between 6-10) to the average pseudo-density design vector ρ _l , the microstructure can be constructed. Expressed as:

Among them, P _l is the p-norm approximation of the average pseudo-density design vector ρ _l , and P _max is the maximum constraint based on the p-norm approximation, which is an empirical parameter, and its value range is 0≤P _max ≤1;

Step 6. Then for the component design variables μ ^K in the multi-component design vector (μ ¹ , μ ² , ...., μ ^k ,...., μ ^K ), k=1, 2,..., K, using the two-step smoothing and projection method shown in the figure below to model multi-component and multi-component entity interfaces (connecting entity interface, entity connecting interface), in which the first step of smoothing and projection obtains the component variables m ^k of each point in the design domain ∈{0, 1}, the second step is smoothing and projection to obtain the physical interface of each component.

其取值范围为

然后再采用DMO(Discrete MaterialOptimization)投影方法，获得设计域内的组件变量m^k∈{0，1}，具体处理公式如下：The specific steps of the first step of smooth projection are: use the Helmhertz partial differential equation to smooth the variable μ ^k to obtain the smoothed variable

Its value range is

Then, the DMO (Discrete Material Optimization) projection method is used to obtain the component variables m ^k ∈ {0, 1} in the design domain. The specific processing formula is as follows:

Among them, r _m is the multi-component interface width control parameter, and its value is set according to the width of the connection interface required by the design, which is an empirical parameter; P _m is the penalty coefficient, which is an empirical parameter, generally 6-15, which can make the projection The latter multi-component variable m ^k basically takes the value of 0 or 1. When the node belongs to the kth component, m ^k =1, and at the same time m ⁱ =0, i∈{1,2,....,K}, and i ≠k.

其取值范围为

然后采用阶跃投影函数获得设计域内取值为0或1的参数ω^k，然后通过简单的乘积方法即可构建第k个组件的实体界面M^k，进一步可估计可加工功能梯度组件C^k，相关公式如下所示：The specific steps of the second step of the smooth projection are: using the Helmhertz partial differential equation to smooth the multi-component variable m ^k after the projection, and obtain the smoothed variable

Its value range is

Then the step projection function is used to obtain the parameter ω ^k with a value of 0 or 1 in the design domain, and then the solid interface M ^k of the kth component can be constructed by a simple product method, and the machinable functional gradient component C ^k can be estimated further, The relevant formulas are as follows:

M ^k =(1-m ^k )w ^k

C ^k =ρm ^k +g(ρ _l )(1-m ^k )ω ^k

为第k个组件的界面宽度控制参数；β，η为阶跃投影函数控制参数，β一般取8-16，主要用于控制阶跃投影函数0-1过度部分的陡峭程度，η是用于控制阶跃投影函数值等于0.5时对应的

的中心位置的横轴坐标值；g(ρ_l)为线性拟合或者样条插值函数，保证多组件实体界面连续，且物理性能跟实体组件部分基本一致。Among them, tanh( ) represents the hyperbolic tangent function;

is the interface width control parameter of the kth component; β, η are the control parameters of the step projection function, β is generally 8-16, mainly used to control the steepness of the transition part of the step projection function 0-1, η is used for When the control step projection function value is equal to 0.5, the corresponding

The horizontal axis coordinate value of the center position of ; g(ρ _l ) is a linear fitting or spline interpolation function, which ensures that the multi-component entity interface is continuous, and the physical performance is basically the same as that of the entity component part.

其中，

和

分别为第i个和第j个组件的界面宽度控制参数，i和j为相邻两个组件编号，i，j＝1，2，....，K，既可以在相邻两个组件界面模型部分产生重叠，重叠部分即为多组件装配连接部分，并通过设置不同物理属性信息，实现不同装配方式的物理建模，实现微结构设计制造一体化设计，多组件装配连接部分物理模型如图3所示，具体建模公式如下：Step 7: By reasonably setting the multi-component interface width control parameter r _m and the single-component interface width control parameter _rs , to ensure

in,

and

are the interface width control parameters of the i-th and j-th components, respectively, i and j are the numbers of two adjacent components, i, j=1, 2, ...., K, both adjacent two components can be The interface model part overlaps, and the overlapping part is the multi-component assembly connection part. By setting different physical attribute information, the physical modeling of different assembly methods is realized, and the integrated design of microstructure design and manufacturing is realized. The physical model of the multi-component assembly connection part is as follows: As shown in Figure 3, the specific modeling formula is as follows:

为第一步和第二步光顺投影之后多组件之间界面具体尺寸参数，

和

分别为第一步和第二步光顺投影之后第i个和第j个组件的实体界面具体尺寸参数。in,

are the specific size parameters of the interface between multiple components after the first and second steps of smooth projection,

and

are the specific size parameters of the entity interface of the i-th and j-th components after the first and second steps of smooth projection, respectively.

By setting the physical property D ₁ of the solid component part and the physical property D ₂ of the connecting part (gluing, welding and riveting, etc.) Mathematical expression I _ij showing the physical properties of the left and right gray and dark gray solid component parts to the middle connecting part.

Step 8. Microstructure product topology optimization mainly builds the multi-component microstructure product model, the multi-component entity interface and the multi-component assembly connection part model; it is necessary to construct the physical properties of each part, and then simulate the component performance and calculation sensitivity through finite element physical simulation. , and then optimize.

In this example, structural static analysis is performed, and the physical properties are taken as Young's modulus.

The physical properties of the solid component part of the kth component, that is, Young's modulus can be expressed as:

Among them, E is Young's modulus, which is valued according to the specific material properties, P is the density penalty coefficient, which is an empirical parameter, and the general value is 3;

The physical properties of the solid interface of the kth component, i.e. Young's modulus, can be expressed as:

The physical properties of the assembly connection part of the i-th component and the j-th component, that is, the Young's modulus can be expressed as:

Among them, E _joint is the physical property of the connection part (Young's modulus), and the designer takes the value according to the physical property of the actual connection process;

Therefore, for each point in the design domain, its total physical property (Young's modulus) can be expressed as:

Step 9, the above steps to step 8, namely the microstructure product topology optimization construction model and key steps, according to the above constructed model, define the objective function as the minimum flexibility function, the constraints include the material use volume ratio constraint, the maximum size of each component Constraints, local average density maximum constraints, and multi-component solid interface and connecting part volume constraints, establish the following optimization objectives:

为第k个组件的最大球形包围盒半径，具体可以表达为：Among them, S is the total stiffness matrix of the structure (each element has a stiffness matrix, which is organized together to form the overall stiffness matrix of the structure); U is the node displacement vector, whose dimension is equal to the number of nodes, and the element of each dimension corresponds to the displacement of a node ;F is the node equivalent load vector, its dimension is equal to the number of nodes, the elements of each dimension correspond to the equivalent load of a node,

is the radius of the largest spherical bounding box of the kth component, which can be expressed as:

Among them, _r is the radius, that is, the distance from each node on the kth component to the center of the component, and rc is the center coordinate of the kth component, which can be expressed as:

采用p范数R_a近似：Since the radius of the spherical bounding box expressed by the maximum value cannot be differentiated, the

Approximate using the p-norm _Ra :

That is, replace the constraint condition with R _a ≤ R _max

The volume fraction V for multi-component materials can be expressed as:

V=∫ _Ω ∑ρm ^k dΩ

The multi-component solid interface and the connected part volume C can be expressed as:

Step 10. Set each parameter value in the objective function and its constraints (including the initial value of φ, μ ^k , K, V ⁰ , R _max , P _max , C ⁰ , D ₁ , D ₂ , rm , _{rs s} etc.), the node displacement vector U is obtained by finite element analysis, and the objective function f is solved by the gradient descent method, and the final result is obtained by iterative convergence.

Example 2:

The present embodiment discloses a multi-component processing method for a microstructure product, comprising the following steps:

Step i, adopt the microstructure design method of multi-component topology optimization in embodiment 1, solve φ, the optimal value of μ ^k , and its corresponding pseudo-density design variable ρ, variable m ^k , average pseudo-density design variable ρ _l and parameter ω ^k ;

Step ii, obtain the mathematical model of each component constituting the microstructure according to the following formula:

C ^k =ρm ^k +g(ρ _l )(1-m ^k )ω ^k

Among them, C ^k represents the kth component that constitutes the microstructure;

Step iii, using processing equipment to process all components according to the mathematical model of each component of the microstructure;

Step iv, assembling each component to obtain a microstructure product.

Example 3:

This embodiment discloses a multi-component topology optimization design system for microstructure products, including a memory and a processor, wherein the memory stores a computer program, and when the computer program is executed by the processor, the processor realizes the The multi-component topology optimization design method of the microstructure product in the above embodiment 1.

Example 4:

This embodiment discloses a computer-readable storage medium on which a computer program is stored, and when the computer program is executed by a processor, implements the multi-component topology optimization design method for a microstructure product in the foregoing embodiment 1.

Example 5:

This embodiment discloses a multi-component processing system for microstructure products, including a memory and a processor, wherein the memory stores a computer program, and also includes processing equipment;

When the computer program is executed by the processor, the processor enables the processor to implement steps i to ii in the method of Embodiment 2 above;

The processing equipment processes all the components according to the mathematical model of each component of the microstructure obtained by the processor;

The individual components are assembled to obtain a microstructured product.

Example 6:

This embodiment realizes the topology optimization design of the microstructure product—the cantilever beam. The traditional structural design method of cantilever beam is to obtain several large members through topology optimization. In this embodiment, the cantilever beam obtained by the topology optimization design is composed of microstructures.

Figure 4 shows the design domain and boundary conditions of topology optimization in this embodiment. Cantilever beam length w=2000mm, height h=1000mm. The left end is fixed and restrained, and the concentrated force load F=1N is applied to the lower right end, and the direction is downward. Selected material Young's modulus E = 1Pa, Poisson's ratio of 0.3.

The design domain is defined as Ω. In the process of solving by the finite element analysis method, the design domain Ω is discretized into 20,000 square elements with r _e = 10mm*10mm, and the vertex of the discrete element square is the node. Initialize the pseudo-density design variable φ=0, in the multi-component design vector (μ ¹ , μ ² , ...., μ ^k , ...., μ ^K ), μ ^k = 0.5, k = 1, 2, . .., K. Set the upper limit value of material usage volume fraction ratio V ⁰ =0.5, the upper limit value of each component size R _max =0.55, the upper limit value of local average density P _max =0.6, and the upper limit value C ⁰ of multi-component solid interface and connection part volume =0.1, set D1= ₁ , D2=0.25, K= ₂ ; rm= _3re , _rs = _{1.75re .} This embodiment is an example of dimensionless design, so each parameter has no unit, and the specific application will take the unit according to the actual situation. Build a topology optimization model:

The finite element analysis was carried out, and the gradient descent optimization algorithm was used to iterate 600 steps, and the topology optimization results of the microstructure cantilever beam composed of two components were converged, as shown in Figure 5.

Example 7:

The difference between this embodiment and Embodiment 6 is that D ₂ =0.75 is set, the strong connection of assembly is simulated, and finite element analysis is performed, and the convergence result of 600 steps of gradient descent iterative optimization is shown in FIG. 6 .

Example 8:

The difference between this embodiment and Embodiment 6 is that D ₂ =0.5 is set to simulate the medium connection strength of the assembly, R _max =0.40 is set, rm = _4re , _rs = _{2.25re ;} K=3 is set to construct 3 Component's microstructure product topology optimization model:

Finite element analysis was carried out, and gradient descent was used for iterative optimization for 600 steps, and the optimization and convergence of the 3-component microstructure product topology optimization results are shown in Figure 7.

Example 9:

The difference between this embodiment and Embodiment 8 is that R _max =0.55 is set;

Finite element analysis was carried out, and gradient descent was used for iterative optimization for 600 steps, and the optimization convergence was achieved until the final optimization was a 2-component microstructure product. The results are shown in Figure 8.

Claims (9)

1. A multi-component topology optimization design method of a microstructure product is characterized by comprising the following steps:

step 1, setting a design domain of topology optimization to be omega; constructing a pseudo-density design variable phi, wherein the phi is a continuous function in the design domain omega, and the value of phi at each position in the design domain omega respectively represents whether materials are arranged at the corresponding position in the design domain omega, and the phi is more than or equal to-1 and less than or equal to 1; construction of a Multi-component design vector (μ)¹，μ²，....，μ^k，....，μ^K) In which μ^kDenotes the kth component design variable, μ^kTo design a continuous function within the domain omega, mu^kThe value of each position in the design domain omega respectively represents the possibility that the corresponding position in the design domain omega belongs to the kth component, and mu is more than or equal to 0^k1, K is equal to or less than 1, 2, K is the maximum number of components for dividing the design domain;

step 2, constructing the following objective function:

wherein c (phi) is a compliance function, and S is a structural total stiffness matrix; u is a node displacement vector, the dimensionality of the node displacement vector is equal to the number of nodes, and an element of each dimensionality corresponds to the displacement of one node; f is a node equivalent load vector, the dimensionality of the node equivalent load vector is equal to the number of nodes, and an element of each dimensionality corresponds to the equivalent load of one node;

v is the material volume ratio; v⁰Using a volume fraction upper limit for the material;

the largest spherical bounding box radius for the kth assembly; r_maxSetting the upper limit value of the size of each component according to the size constraint of the machining method;

P_ldesign variable ρ for average pseudo-density_lApproximation of P-norm of (1), P_maxIs the upper limit value of local average density, and is not less than 0 and not more than P_max≤1；

C is the volume of the solid interface and connecting part of the multi-component⁰The volume upper limit value of the multi-component solid interface and the connecting part is set;

step 3, solving an objective function; setting the assembly connection mode and the physical property of the assembly connection part for each group of phi and mu^kTaking values, and acquiring corresponding U, S, F and a corresponding objective function value by adopting a finite element method; iterating to obtain phi, mu^kThe optimal value of (2);

step 4, according to phi and mu^kDetermining whether the material is arranged at each position in the design domain omega, and forming a microstructure product model by all parts of the material arranged in the design domain omega.

2. The method for the topological optimized design of multiple components of a microstructure product according to claim 1, wherein the material usage volume fraction ratio V ═ jjj ^ j_Ω∑_kρm^kdΩ；

Wherein rho is a pseudo density design variable which basically has only two values of 0 and 1 in a design domain, and is obtained through the following steps:

firstly, adopting a Helmholtz partial differential equation to design a variable phi through light-smoothening pseudo density, wherein the formula is as follows:

in the formula

In order to be a gradient operator, the method comprises the following steps,

designing variables for the smooth pseudo-density; r is_ρThe minimum geometric dimension of the topological optimization structure can be controlled at the same time due to the smooth radius;

then, a step projection function is used

Obtaining a pseudo-density design variable rho, wherein the formula is as follows:

wherein h is a control parameter of the continuous transition part of the step projection;

m^kfor designing the k component variable, m, of each cell in the domain^kE {0, 1}, obtained by the following steps:

firstly, the partial differential equation of Helmholtz is used to make variable mu^kPerforming fairing treatment to obtain the smoothed variable

The formula is as follows:

then, using DMO projection method, m is obtained^kThe specific processing formula is as follows:

wherein r is_mControlling parameters for the multi-component interface width; p_mIs a penalty factor.

3. The method of claim 2, wherein the method comprises designing a multi-component topology of the microstructured product according to the method of claim 2

By the use of R_aIn the approximation that the difference between the first and second values,

where r is the radius, i.e. the distance of each node on the kth element from the centre of the element, r_cIs the center coordinate of the kth component, p is a constant.

4. The method of claim 2, wherein P is P_lSolving by the following steps:

firstly, processing a pseudo-density design variable rho through a partial differential equation of Helmholtz with variable radius to obtain an average pseudo-density design variable rho in a local field_lThe formula is as follows:

wherein r is_lIs an average density neighborhood control variable;

then, an average pseudo-density design variable ρ is calculated_lBased on p-norm approximation:

wherein, P_lDesign variable ρ for average pseudo-density_lApproximate p-norm of (d).

5. The method of claim 2, wherein the multi-component physical interface and the volume of the connecting portion C are calculated by the following formula:

wherein g (ρ)_l) Linear fitting or spline interpolation function; m^kA physical interface for the kth component obtained by:

firstly, fairing processing is carried out on variable mk by adopting Helmholtz partial differential equation to obtain the fairing variable

The value range is

The formula is as follows:

in the formula,

controlling parameters for the interface width of the kth element;

then, a step projection function is adopted to obtain a parameter omega with the value of 0 or 1 in the design domain^k：

Wherein, tanh (DEG) represents a hyperbolic tangent function, and β and η are control parameters of the step projection function;

and then constructing an entity interface M of the kth component by a product method^k：

M^k＝(1-m^k)w^k；

I_ijThe physical property of the entity component part of the ith and jth components to the connection part between the entity component part and the jth component is represented by the following mathematical expression:

wherein D is₁And D₂Physical attributes of the entity component part and the connection part are respectively.

6. A method for processing a plurality of components of a microstructure product is characterized by comprising the following steps:

step i, solving for φ, μ using the method of claim 5^kAnd the corresponding pseudo density design variable rho and the variable m^kAverage pseudo density design variable ρ_lAnd parameter omega^k；

Step ii, obtaining a mathematical model of each component forming the microstructure according to the following formula:

C^k＝ρm^k+g(ρ_l)(1-m^k)ω^k

wherein, C^kRepresenting the kth component constituting the microstructure;

step iii, processing all components by adopting processing equipment according to the mathematical model of each component of the microstructure;

and iv, assembling each component to obtain the microstructure product.

7. A system for the optimized design of a multi-component topology of a microstructure product, comprising a memory and a processor, wherein the memory stores a computer program, and wherein the computer program, when executed by the processor, causes the processor to carry out the method according to any one of claims 1 to 5.

8. A computer-readable storage medium, on which a computer program is stored, which, when being executed by a processor, carries out the method according to any one of claims 1 to 5.

9. The multi-component processing system for the microstructure product comprises a memory and a processor, wherein the memory is stored with a computer program, and is characterized by further comprising processing equipment;

the computer program, when executed by the processor, causes the processor to carry out steps i to ii of the method of claim 7;

the processing equipment processes all components according to the mathematical model of each component of the microstructure obtained by the processor;

and assembling the components to obtain the microstructure product.

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