CN111027150A - Multi-component topology optimization design and processing method and system for microstructure product - Google Patents
Multi-component topology optimization design and processing method and system for microstructure product Download PDFInfo
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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,….,μk,….,μK) 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 mukThe optimal value of (2); according to phi, mukDetermining 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
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 and processing method and system for a microstructure product.
Background
The microstructure not only has the characteristic of light weight and excellent performance, but also can realize a plurality of functions of negative Poisson's ratio, zero thermal expansion, high heat dissipation ratio and performance ratio, frequency band sound absorption performance, light guidance and the like, so that the microstructure has great industrialized application requirements in the fields of automobiles, ships, trains, aerospace and the like.
The existing method for designing microstructure products (products formed by microstructures, including large components formed by microstructures) mainly comprises the steps of designing microstructure units, adding periodic boundary conditions, adopting finite element simulation analysis, and finally realizing the microstructure products in a mode of tiling the microstructure units. The method is an trial design method, different microstructure units need to be repeatedly tried and designed, and a microstructure product with better performance and meeting performance requirements can be obtained through repeated experiments, so that the labor and material consumption is huge. In recent years, topological optimization methods have been applied to the field of design of microstructure products. However, the existing microstructure product design method based on topology optimization has the problems that the connection of microstructure units is not smooth, and the like, and a large amount of post-processing is needed to ensure the connection smoothness between the microstructure units, but the physical performance of the optimized structure is inevitably lost in the post-processing process. Moreover, because the microstructure products designed by topology optimization are different in form, the traditional processing method can not basically realize the processing of the microstructure products designed by topology optimization or has huge processing cost, so that the existing microstructure products are mostly processed by 3D printing. However, the existing 3D printers have size limitations, and especially for metal materials, there are also problems of deformation and fracture due to excessive residual stress during the printing process of large components (large-sized components), so that processing large-sized microstructure products becomes a current research difficulty, and is also a key for restricting the application of microstructures in complex product large components in the fields of automobiles, ships, trains, aerospace and the like.
The multi-component topological optimization method in the large component structure design method is to optimize the large component into a plurality of small components meeting the size constraint of a processing method (such as the maximum processing size of a 3D printer), and then assemble the plurality of small components to obtain the large component. However, the existing multi-component topology optimization method mainly aims at the optimization of solid components, and is directly applied to large components composed of microstructures, and the following problems can exist: the microstructure may be divided into a plurality of members during the process, thereby causing a great challenge to the microstructure unit connection portions of the plurality of members. Meanwhile, the physical attributes of the component connecting parts and the manufacturing process constraints are not considered in the current multi-component topology optimization method based on the gradient method, but the traditional multi-component topology optimization method adopting the genetic algorithm considers the physical attributes of the component connecting parts, but the calculation efficiency is very low, and the requirement of designing variable optimization of thousands of components or even hundreds of thousands of components cannot be met.
Therefore, in order to meet the requirement of industrial application of the microstructure in large components, a design and manufacturing integrated microstructure product topology optimization design method is needed.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a method and a system for topological optimization design and processing of a plurality of components of a microstructure product, aiming at the defects of the prior art, so that the topological optimization design and processing of the microstructure product can be realized, and the precision is high.
A multi-component topological optimization design method for a microstructure product comprises the following steps:
step 2, constructing the following objective function:
wherein c (phi) is a compliance function, S is a structure total stiffness matrix (each unit in a design domain omega has a stiffness matrix, and the structure total stiffness matrix is formed by organizing the units together); u is a node displacement vector, the dimensionality of the node displacement vector is equal to that of a node, 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 (maximum constraint);
Pldesign variable ρ for average pseudo-densitylApproximation of P-norm of (1), PmaxFor the local mean density upper limit (maximum constraint), P is greater than or equal to 0max≤1；
C is the volume of the solid interface and connecting part of the multi-component0Physically interfacing and connecting multiple componentsPartial volume upper limit (maximum constraint);
each upper limit value is manually set according to needs/experience;
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 mukTaking values, and obtaining corresponding U, S, F and corresponding objective function values by adopting a finite element analysis method (dispersing a design domain omega into a plurality of units, wherein the top point of each unit is a node); iterative solution to obtain phi and mukThe optimal value of (2);
step 4, according to phi and mukDetermining 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 in the processing (manufacturing) process of the microstructure product is considered, the method is a design and manufacturing integrated method, and the high-precision processing requirement of a large component of a complex product can be met.
Further, the material uses a volume fraction ratio V ═ -Ω∑kρmkd Ω, i.e. ΣkρmkIntegration over the entire design domain;
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 isρTo smooth the radius while being able to control the topologyOptimizing the minimum geometry of the structure, rρThe parameter is an empirical parameter, and generally takes a value larger than the minimum processing precision of a processing mode;
then, a step projection function is used
Obtaining a pseudo-density design variable rho of which the value is basically only 0 and 1 in a design domain omega, wherein the formula is as follows:
in the formula, h is a control parameter of a continuous transition part of the step projection, is an empirical parameter and generally takes a value of 0.5;
mkfor designing the k component variable, m, of each cell in the domainkE {0, 1}, obtained by the following steps:
firstly, the partial differential equation of Helmholtz is used to make variable mukPerforming fairing treatment to obtain the smoothed variable
The value range is
The formula is as follows:
then, using DMO projection method, m is obtainedkThe specific processing formula is as follows:
wherein r ismControlling parameters for the width of the interface of the multi-component, namely empirical parameters, wherein the values are set according to the width of the connecting interface of the component required by design; pmThe penalty coefficient is an empirical parameter and generally takes a value between 6 and 15;
the above steps enable the variable m obtained after projectionkThe basic value is 0 or 1; when a certain position in the design domain omega belongs to the kth component, the corresponding m k1 while it corresponds to mi0, i ∈ {1, 2.., K }, and i ≠ K.
Further, the
By the use of RaIn 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, rcThe k-th component's center coordinate can be expressed as:
p is an empirical parameter, typically a number between 6 and 10.
Further, said PlSolving 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 fieldlThe formula is as follows:
wherein r islIs an average density neighborhood control variable with the value larger than rρ；
Then, an average pseudo-density design variable ρ is calculatedlBased on p-norm approximation:
wherein, PlDesign variable ρ for average pseudo-densitylApproximate p-norm of (d).
Further, the multi-component solid interface and connecting portion volume C is calculated by the following formula:
wherein g (ρ)l) Linear fitting or spline interpolation function; mkA physical interface for the kth component obtained by:
firstly, the partial differential equation of Helmholtz is used to the variable mkPerforming fairing treatment to obtain the smoothed variable
The value range is
The formula is as follows:
in the formula, rs kControlling parameters for the interface width of the kth assembly, wherein the parameters are empirical parameters, and values of the empirical parameters are set according to the width of the connection interface of the kth assembly;
then, a step projection function is adopted to obtain a parameter omega with the value of 0 or 1 in the design domaink：
Wherein, tanh (DEG) represents hyperbolic tangent function, β and η are control parameters of the step projection function, β is generally 8-16 and is mainly used for controlling the steepness of the 0-1 transition part of the step projection function, η is used for controlling the corresponding step projection function value to be equal to 0.5
A value;
then, the entity interface M of the kth component can be constructed by a simple product methodk：
Mk＝(1-mk)wk；
IijThe 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. I isijNamely a multi-component assembly connection physical model.
The invention also provides a multi-component processing method of the microstructure product, which comprises the following steps:
step i, adopting the multi-component topological optimization microstructure design method to solve phi and mukAnd the corresponding pseudo density design variable rho and the variable mkAverage pseudo density design variable ρlAnd parameter omegak；
Step ii, obtaining a mathematical model of each component forming the microstructure according to the following formula:
Ck＝ρmk+g(ρl)(1-mk)ωk
wherein, CkRepresenting 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.
The invention also provides a multi-component topology optimization design system of the microstructure product, which comprises a memory and a processor, wherein the memory stores a computer program, and the system is characterized in that when the computer program is executed by the processor, the processor realizes any one of the above multi-component topology optimization design methods of the microstructure product.
The invention also provides a computer readable storage medium, on which a computer program is stored, wherein the computer program is executed by a processor to implement any one of the above methods for the optimized design of the multi-component topology of the microstructure product.
The invention also provides a multi-component processing system of the microstructure product, which comprises a memory and a processor, wherein the memory stores a computer program and is characterized by also comprising processing equipment;
the computer program, when executed by the processor, causes the processor to perform steps i-ii of the method;
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.
The invention discloses a multi-component topological optimization design and processing method and a system of a microstructure product, which mainly comprise the steps of functional gradient microstructure optimization design (namely, pseudo-density design variable optimization), multi-component optimization segmentation (namely, multi-component design vector optimization), multi-component connection interface physical model construction and optimization and the like. The model can be mainly divided into three levels, wherein the first level is a basic design parameter and comprises a pseudo density design variable (density variable) and a component design variable (component variable); the second layer is an intermediate variable and comprises a multi-component entity interface, a multi-component assembly connection physical modeling and the like; the third layer of structure design output, including the multi-component that can be processed and the final assembly result, is shown in fig. 1, where K is 3 as an example. Through topology optimization, the obtained optimization result is a microstructure product in a design domain, such as a black part geometric structure shown in fig. 1, wherein a black part is a part for arranging materials, a white part is empty, and no materials are arranged.
Has the advantages that:
the invention provides a multi-component topology optimization design and processing method and system of a microstructure product, and a computer readable storage medium, which can realize the topology optimization design of the microstructure product and have high precision; and has the following advantages:
(1) the microstructure in the microstructure product is designed by adopting a topological optimization method, and compared with the existing trial design method of the microstructure, the method has the advantages that the consumption of manpower and material resources is low;
(2) by a variable radius Helmholtz partial differential equation fairing method and variable density topological optimization, the topological optimization design of fairing connection (fairing connection between microstructure units and between the microstructure units and assemblies) of the microstructure units and the assemblies is realized; the microstructure product designed by the invention is composed of a functional gradient microstructure (functional gradient microstructure), and comprises a plurality of microstructure units with similar sizes and shapes, wherein the sizes and the shapes of the microstructure units at different positions are slightly different so as to meet the performance requirements of the corresponding positions;
(3) by adopting a cladding modeling method (two-step fairing and projection method), a multi-component entity interface of the microstructure product is constructed, a multi-component assembly connection physical model is further constructed, multi-scale fairing connection between a microstructure unit and a component can be ensured, and meanwhile, simulation of assembly modes of the components including gluing, welding, riveting and the like is realized through assembly connection modeling, so that design and manufacturing integration of the microstructure product is realized;
(4) a post-processing stage in the traditional microstructure topology optimization design is not needed, and the physical performance loss of the post-processing stage is avoided;
(5) in the design process of the microstructure product, the size constraint of the processing method in the processing (manufacturing) process of the microstructure product is considered, and the multi-component topological optimization design method is adopted, so that the design and manufacturing integration can be realized, the size constraint of additive manufacturing equipment can be met, and the high-precision processing requirement of a large component of a complex product can be met; the method has the advantages that the processing of large-scale additive manufacturing equipment is not needed, a large amount of processing cost is avoided, the risks of defect generation and residual stress deformation in the additive manufacturing process of the large component are greatly reduced, and the high-precision high-quality complex product large component can be constructed through the existing small additive manufacturing equipment or the traditional processing mode. 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) The microstructure is automatically generated in the topological optimization process based on the pseudo-density design variable without a fixed unit type, the constraint of limited search space caused by the fact that the initial form of the microstructure unit is fixed by the traditional design and optimization method is avoided, and the minimum flexibility of the optimized microstructure product is very close to the topological optimization result of the entity structure of the variable density method.
Drawings
FIG. 1 is a topological optimization model framework of a microstructure product (components 3 are an example);
FIG. 2 is a schematic diagram of a two-step fairing and projection method for building component physical interfaces;
FIG. 3 is a geometric and physical modeling of inter-component connections;
FIG. 4 shows the cantilever topology optimization design domain and boundary conditions
FIG. 5 shows the topological optimization of the microstructure product of the assembly weak link 2 component, wherein FIG. 5(a) shows component variables 1, FIG. 5(b) shows component variables 2, FIG. 5(c) shows the optimized machinable component 1, FIG. 5(d) shows the optimized machinable component 2, FIG. 5(e) shows the microstructure-optimized assembly result, and FIG. 5(f) shows the multi-component solid interface and the assembly link portion (gray portion is material, black portion is empty)
FIG. 6 is a topological optimization of a microstructure product of an assembly strong connection 2 component, wherein FIG. 6(a) is a component variable 1, FIG. 6(b) is a component variable 2, FIG. 6(c) is an optimized machinable component 1, FIG. 6(d) is an optimized machinable component 2, and FIG. 6(e) is a microstructure optimized assembly result, wherein FIG. 6(f) is a multi-component solid interface and an assembly connection portion (gray portion is material, black portion is empty)
FIG. 7 is a topological optimization of a 3-component microstructure product with medium joining strength and small machining dimension constraints in assembly, where FIG. 7(a) is component variable 1, FIG. 7(b) is optimized machinable component 1, FIG. 7(c) is component variable 2, FIG. 7(d) is optimized machinable component 2, FIG. 7(e) is component variable 3, FIG. 7(f) is optimized machinable component 3, FIG. 7(g) is a microstructure optimized assembly result, and FIG. 7(h) is a multi-component solid interface and an assembly joint
Fig. 8 is a topological optimization of a microstructure product of 3 components with medium joining strength and large machining dimension constraint in assembly, where fig. 8(a) is component variable 1, fig. 8(b) is optimized machinable component 1, fig. 8(c) is component variable 2, fig. 8(d) is optimized machinable component 2, fig. 8(e) is component variable 3, fig. 8(f) is optimized machinable component 3, fig. 8(g) is a microstructure optimized assembly result, and fig. 8(h) is a multi-component solid interface and an assembly joining part.
Detailed Description
The invention is described in detail below with reference to the figures and the specific examples.
Example 1:
the embodiment discloses a multi-component topology optimization design method of a microstructure product, which comprises the following steps:
step one, setting a design domain of topology optimization to be omega; constructing a pseudo-density design variable (topological geometric design variable) phi which is a continuous function in a design domain omega, wherein 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 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，....，μk，....，μK) In which μkDenotes the kth component design variable, μkTo design a continuous function within the domain omega, mukThe 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 0k1, K1, 2, K being the maximum number of components (artificial setting) into which the design domain is divided; in the multi-component design vector corresponding to each position in the omega design domain obtained through optimization, only one element is 1, and other elements are 0; if the final optimized multi-component design vector corresponding to a certain position is obtained, mu k1, the position belongs to the k component;
step two, adopting a Helmholtz partial differential equation light-smoothening pseudo-density design vector phi to solve the checkerboard problem in the optimization process, wherein the formula is as follows:
wherein,
in order to be a gradient operator, the method comprises the following steps,
designing a vector for the smoothed pseudo-density; r isρTo smooth the radius while controlling the minimum geometry of the topology optimization structure, rρThe parameter is an empirical parameter, and generally takes a value larger than the minimum processing precision of a processing mode;
step three, designing vector of pseudo density after fairing
Is [ -1, 1 [ ]]Variation of range, then using step projection function
Obtaining a pseudo-density design variable rho basically only having two values of 0 and 1 in a design domain, wherein the rho comprises two values of 0 and 1 (0 represents no material and 1 represents material), and a narrow transition band between the two values to realize smooth connection, so that the pseudo-density design variable rho is optimized by a gradient method and is convenient to process and manufacture, and the specific formula is as follows:
wherein h is a control parameter of the continuous transition part of the step projection, is an empirical parameter, and generally takes a value of 0.5.
Step four, processing the pseudo density design vector rho through a partial differential equation of Helmholtz with variable radius to obtain an average pseudo density design vector rho in the local fieldlThe method comprises the following steps:
wherein r islIs an average density neighborhood control variable, is an empirical parameter, and has a value greater than rρ。
Step five, designing a vector rho by the average pseudo densitylAdding a maximum value constraint P based on P-norm (P is an empirical parameter, typically a number between 6 and 10) approximationmaxI.e. byA microstructure can be constructed, which can be specifically expressed as:
wherein, PlDesign vector ρ for average pseudo-densitylApproximation of P-norm of (1), PmaxIs a maximum value constraint based on P norm approximation, is an empirical parameter, and has a value range of P being more than or equal to 0max≤1；
Step six, then designing vectors (mu) for the multiple components1，μ2，....，μk，....，μK) Component design variable μ inKK, modeling multi-component and multi-component solid interfaces (connecting solid interfaces, solid connecting interfaces) using a two-step ray sum projection method as shown in the following figure, wherein the first step ray sum projection obtains component variables m for each point in the design domainkE {0, 1}, and the second step of optical order and projection obtains the entity interface of each component.
The first step of the fairing projection comprises the following specific steps: using partial differential equation of Helmholtz to variable mukPerforming fairing treatment to obtain the smoothed variable
The value range is
Then, using DMO (discrete material optimization) projection method to obtain component variable m in design domainkE {0, 1}, and the specific processing formula is as follows:
wherein r ismControl parameters for the width of the multi-component interface, the values of which are determined according to the design requirements of the connection interfaceSetting the width as an empirical parameter; pmThe penalty coefficient is an empirical parameter, generally takes 6-15, and can enable the multi-component variable m after projectionkBasically takes the value 0 or 1, m is when the node belongs to the kth component k1 while mi0, i ∈ {1, 2.., K }, and i ≠ K.
The second step of the fairing projection comprises the following specific steps: using Helmholtz partial differential equation to project multi-component variable mkPerforming fairing treatment to obtain the smoothed variable
The value range is
Then, a step projection function is adopted to obtain a parameter omega with the value of 0 or 1 in a design domainkThen, the entity interface M of the kth component can be constructed by a simple product methodkFurther, a workable functional gradient component C can be estimatedkThe correlation formula is as follows:
Mk＝(1-mk)wk
Ck＝ρmk+g(ρl)(1-mk)ωk
wherein, tanh (·) represents a hyperbolic tangent function;
β is control parameter of interface width of kth module, β is control parameter of step projection function, which is generally 8-16, and is mainly used for controlling steepness of 0-1 transition part of step projection function, η is used for controlling corresponding step projection function value equal to 0.5
A horizontal axis coordinate value of the center position of (a); g (rho)l) The method is a linear fitting function or a spline interpolation function, ensures that the multi-component solid interface is continuous, and the physical properties are basically consistent with those of the solid component part.
Step seven: by reasonably setting the control parameter r of the width of the interface of a plurality of componentsmAnd a single-component interface width control parameter rsTo ensure
Wherein,
and
the interface width control parameters of the ith and jth components are respectively, i and j are serial numbers of two adjacent components, i and j are 1, 2, and K, so that the two adjacent component interface model parts can be overlapped, the overlapped part is a multi-component assembly connecting part, physical modeling of different assembly modes is realized by setting different physical attribute information, and the integrated design of microstructure design and manufacture is realized, wherein the physical model of the multi-component assembly connecting part is shown in figure 3, and the specific modeling formula is as follows:
wherein,
the specific size parameters of the interface between the multiple components after the first step and the second step of the fairing projection,
and
the concrete dimension parameters of the entity interface of the ith assembly and the jth assembly after the first step and the second step of the fairing projection are respectively.
By setting physical property D of entity component part1And physical properties D of the joint (gluing, welding, riveting, etc.)2(the physical properties may be various physical properties such as Young's modulus, specific heat capacity, etc.) the mathematical expression I of the physical properties of the solid component parts to the intermediate connection parts of the left and right gray and dark gray shown in FIG. 3 can be realizedij。
Step eight, topological optimization of the microstructure product mainly constructs a multi-component microstructure product model, a multi-component solid interface and a multi-component assembly connecting part model; physical properties of all parts need to be constructed, then the performance and the calculation sensitivity of the component are simulated through finite element physical simulation, and then optimization is carried out.
In this example, structural statics analysis was performed, and the physical property was young's modulus.
The physical property of the solid component part of the kth component, i.e. young's modulus, can be expressed as:
wherein E is Young modulus, and is taken as a value according to specific material properties, P is a density penalty coefficient, is an empirical parameter, and is generally taken as 3;
the physical interface physical property of the kth component, i.e. young's modulus, can be expressed as:
the physical properties of the assembly connecting parts of the ith component and the jth component, namely Young modulus, can be expressed as:
wherein E isjointThe physical property (Young modulus) of the connecting part is obtained by a designer according to the physical property of the actual connecting process;
thus, for each point within the design domain, its total physical property (Young's modulus) can be expressed as:
step nine, the steps to the step eight, namely a topological optimization construction model and a key step of the microstructure product, according to the constructed model, an objective function is defined as the minimum of a compliance function, constraint conditions comprise material use volume ratio constraint, maximum size constraint of each component, local average density maximum value constraint and multi-component entity interface and connecting part volume constraint, and the following optimization objective is established:
wherein S is a structural total stiffness matrix (each unit has a stiffness matrix, and the structural total stiffness matrix is formed by organizing the units together); 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, each dimensionality element corresponds to the equivalent load of one node,
the radius of the maximum spherical bounding box of the kth component can be expressed as:
where r is the radius, i.e., the distance from each node on the kth component to the center of the component, rcThe k-th component's center coordinate can be expressed as:
since the radius of the spherical bounding box expressed by the maximum is not differentiated, in the constraint
Using p-norm RaApproximation:
namely Ra≤RmaxIn alternative constraints
The multi-component material usage volume ratio V can be expressed as:
V＝∫Ω∑ρmkdΩ
the multi-component solid interface and connecting portion volume C can be expressed as:
step ten, setting each parameter value (including phi and mu) in the objective function and the constraint condition thereofkK, V of0、Rmax、Pmax、C0、D1、D2、rm、rsAnd the like), obtaining a node displacement vector U by adopting finite element analysis and calculation, solving an objective function f by adopting a gradient descent method, and iteratively converging to obtain a final result.
Example 2:
the embodiment discloses a method for processing a multi-component of a microstructure product, which comprises the following steps:
step i, adopting the multi-component topology optimization microstructure design method in the embodiment 1 to solve phi and mukAnd the corresponding pseudo density design variable rho and the variable mkAverage pseudo density design variable ρlAnd parameter omegak；
Step ii, obtaining a mathematical model of each component forming the microstructure according to the following formula:
Ck＝ρmk+g(ρl)(1-mk)ωk
wherein, CkRepresenting 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.
Example 3:
the embodiment discloses a multi-component topology optimization design system of a microstructure product, which comprises a memory and a processor, wherein a computer program is stored in the memory, and when the computer program is executed by the processor, the processor is enabled to realize the multi-component topology optimization design method of the microstructure product in the embodiment 1.
Example 4:
this embodiment discloses a computer-readable storage medium, on which a computer program is stored, and the computer program, when executed by a processor, implements the method for multi-component topology optimization design of a microstructure product in embodiment 1 above.
Example 5:
the embodiment discloses a multi-component processing system of a microstructure product, which comprises a memory, a processor, a processing device and a processing device, wherein the memory stores a computer program;
the computer program, when executed by the processor, causes the processor to perform steps i to ii of the method of embodiment 2 above;
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.
Example 6:
the embodiment realizes the topological optimization design of the microstructure product-cantilever beam. The traditional structural design method of the cantilever beam is to obtain a plurality of large components through topological optimization. In the embodiment, the obtained cantilever beam is composed of a microstructure through topology optimization design.
Fig. 4 shows design domains and boundary conditions for topology optimization in this embodiment. The length w of the cantilever beam is 2000mm, and the height h is 1000 mm. The left end is fixedly restrained, and the right lower end applies a concentrated force load F equal to 1N in a downward direction. The Young modulus E of the selected material is 1Pa, and the Poisson ratio is 0.3.
Defining a design domain as omega, and dispersing the design domain omega into 2 ten thousand r in the process of solving by using a finite element analysis methode10mm square cells, the vertices of the discrete cell squares are the nodes. Initializing a pseudo-density design variable phi 0, a multi-component design vector (mu)1，μ2，....，μk，....，μK) Mu ink0.5, K is 1, 2. Upper limit value V of material use volume ratio0Upper limit of each component size R0.5max0.55, local average Density Upper Limit value Pmax0.6, and upper limit of the volume of the multi-component solid interface and the connecting part C0Set D equal to 0.11＝1，D2＝0.25，K＝2；rm＝3re，rs＝1.75re. The embodiment is an example of dimensionless design, so each parameter has no unit, and the specific application is to take the unit according to the actual situation. Establishing a topology optimization model:
finite element analysis is carried out, a gradient descent optimization algorithm is adopted to iterate 600 steps, and a topological optimization result of the microstructure cantilever beam formed by the 2 components is obtained through convergence, as shown in figure 5.
Example 7:
this embodiment is different from embodiment 6 in that D is set2At 0.75, a strong connection was simulated and subjected to finite element analysis, and the convergence results of step 600 of iterative optimization using gradient descent are shown in fig. 6.
Example 8:
this embodiment is different from embodiment 6 in that D is set20.5, simulate moderate connection strength in assembly, set Rmax0.40, rm＝4re，rs＝2.25re(ii) a Setting K to be 3, constructing a microstructure product topology optimization model of 3 components:
finite element analysis is carried out, gradient descent iterative optimization is adopted for 600 steps, and the topological optimization result of the optimized convergence 3 component microstructure product is shown in FIG. 7.
Example 9:
this embodiment differs from embodiment 8 in that R is setmax＝0.55；
Finite element analysis is carried out, gradient descent iterative optimization is adopted for 600 steps, optimization convergence is achieved, and a microstructure product which is finally optimized into 2 components is obtained, so that the method can automatically remove unnecessary components in the design process, the number and the structure of the optimized components are obtained, and the specific result is shown in fig. 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，....，μk，....，μK) In which μkDenotes the kth component design variable, μkTo design a continuous function within the domain omega, mukThe 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 0k1, 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;
Pldesign variable ρ for average pseudo-densitylApproximation of P-norm of (1), PmaxIs the upper limit value of local average density, and is not less than 0 and not more than Pmax≤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 mukTaking values, and acquiring corresponding U, S, F and a corresponding objective function value by adopting a finite element method; iterating to obtain phi, mukThe optimal value of (2);
step 4, according to phi and mukDetermining 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 ^ jΩ∑kρmkdΩ；
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 isρThe 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;
mkfor designing the k component variable, m, of each cell in the domainkE {0, 1}, obtained by the following steps:
firstly, the partial differential equation of Helmholtz is used to make variable mukPerforming fairing treatment to obtain the smoothed variable
The formula is as follows:
then, using DMO projection method, m is obtainedkThe specific processing formula is as follows:
wherein r ismControlling parameters for the multi-component interface width; pmIs 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 RaIn 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, rcIs the center coordinate of the kth component, p is a constant.
4. The method of claim 2, wherein P is PlSolving 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 fieldlThe formula is as follows:
wherein r islIs an average density neighborhood control variable;
then, an average pseudo-density design variable ρ is calculatedlBased on p-norm approximation:
wherein, PlDesign variable ρ for average pseudo-densitylApproximate 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; mkA 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 domaink：
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 methodk：
Mk＝(1-mk)wk；
IijThe 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 5kAnd the corresponding pseudo density design variable rho and the variable mkAverage pseudo density design variable ρlAnd parameter omegak；
Step ii, obtaining a mathematical model of each component forming the microstructure according to the following formula:
Ck＝ρmk+g(ρl)(1-mk)ωk
wherein, CkRepresenting 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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