Sequential topological optimization method for multi-material structure design

Abstract

A sequential topological optimization method for multi-material structure design comprises the steps that firstly, a proxy material model design domain and components are defined, a mobile deformable component method is adopted for topological optimization to determine an optimal topological structure, and component mapping units under mapping of topological description equations are determined according to the topological description equations of all the components of a proxy material model; and selecting and optimizing the material of each component mapping unit in the range of the optimal topological structure by adopting a variable density method so as to obtain a multi-material structure. Topological structure optimization and material selection optimization are connected by utilizing a proxy material model to identify a mapping unit, the change of internal material layout does not influence an external topological structure, and additional calculation amount is not increased. The topological structure is optimized in advance, design variables of material selection are reduced during material selection optimization, and the calculation efficiency can be obviously improved. The method is simple and easy to implement and also has a wide application prospect.

Description

Technical Field

[0001] The invention relates to a topology optimization method for a multi-material structure, in particular to a sequential topology optimization method for multi-material structure design. Background Art

[0002] Compared with single-material structures, multi-phase material structures can make more reasonable use of multiple materials with different properties to jointly bear the structural force to meet different design requirements. At the same time, multi-material structures can further achieve the lightweight level of the structure and provide a broader lightweight design idea. With the rapid development of additive manufacturing technology, it has become increasingly convenient to manufacture multi-material structures at a relatively low cost, so multi-material structure design has received more and more attention from the academic community.

[0003] In order to make more effective use of multi-material structures, it is becoming increasingly urgent to develop efficient multi-material structure design methods. As a method of finding the optimal material layout within a predetermined design domain to meet the structural performance requirements, topology optimization has been widely developed. In recent years, the use of topology optimization methods to achieve multi-material structure design has gradually become a research hotspot. Compared with the single-material topology optimization method that only determines the presence or absence of materials, multi-material topology optimization can be seen as a problem of determining the material state of each point in a design domain, that is, by determining whether there is a material at each point in the design domain and what kind of material exists, the optimal distribution of multiple materials in the structure is achieved. The core content of multi-material topology optimization lies in how to achieve the parameterization of materials in multi-materials. The key issue is to construct the so-called material interpolation mechanism to represent the material property indicator function (such as the density field in the SIMP method, the level set function in the level set method, and the component description function in the MMC method).

[0004] Multi-material structural topology optimization requires not only determining the topology of the structure, but also the type of material in the structure. One method is to optimize the structural topology and material selection simultaneously, which is a global optimization (often requiring multiple material selection variables to be set for each unit). Material information is selected in each iteration step. During the optimization process, the structural topology information and material property information are exchanged in real time, affecting sensitivity calculations and variable updates. This method considers changes in material information when optimizing the topological structure, so the structural optimization accuracy is high and a better multi-material structure can be obtained, but its optimization process is complex and the single iteration calculation scale is large.

[0005] Therefore, the design variables of the existing multi-material topology optimization methods will also increase significantly after the introduction of multi-material parameters, which will have a greater impact on the computational efficiency. And when multiple materials are incorporated into the topological design of the structural system, the optimization model and its corresponding solution process will become more complicated due to the interaction of multiple material phases in the ultra-high-dimensional design space. Summary of the invention

[0006] The technical problem to be solved by the present invention is to overcome the defects of complex multi-material topology optimization process, large single iteration calculation scale and affecting calculation efficiency, and provide a sequential topology optimization method for multi-material structure design.

[0007] The technical solution adopted by the present invention to solve the above technical problems is: a sequential topology optimization method for multi-material structure design, first defining the design domain and components of the proxy material model, using the mobile deformable component method to perform topology optimization to determine the optimal topology structure, and then using the variable density method to optimize the material selection of each unit in the topology structure to obtain the multi-material structure, including the following steps:

[0008] (1) Use the mobile deformable component method to obtain the optimization result of the optimal topological structure and determine the topological description equations of all components of the proxy material model under the optimal topological structure;

[0009] (2) According to the topological description equation, determine the component mapping units under the mapping of the topological description equation, including entity units and boundary units;

[0010] (3) constructing a material selection model, determining the design variables of the variable density method, and selecting the material of each component mapping unit within the range of the optimal topological structure;

[0011] (4) Perform finite element analysis and sensitivity analysis, update the design variables of the variable density method and check the convergence.

[0012] After obtaining the optimal topological structure of the proxy material model by moving the deformable component method, the topological function value of each unit of the proxy material model is determined based on the topological description equation, and the units with unit density greater than 0 are identified and determined as component mapping units under the mapping of the topological description equation.

[0013] The moving deformable component method is used to perform topology optimization based on one material and determine the optimal topology structure. Then the variable density method is used to determine whether the component mapping unit selects the second material to achieve material selection optimization.

[0014] The variable density method is used to optimize the selection of two-phase materials for each component mapping unit, and the material selection model is constructed as follows:

[0015] where ρ e Select density for the material, E1 and E2 are the Young's modulus of the two materials respectively, and p is the penalty coefficient, p ≥ 3.

[0016] The variable density method takes the minimum flexibility of the structure as the objective function, and takes the total volume of the structure and the volume proportion of each material relative to the base material as constraints to optimize the material density selection variables.

[0017] Construct the material selection stiffness matrix of the component mapping unit for finite element analysis. The component mapping unit stiffness matrix can be expressed as:

[0018] The overall stiffness matrix is in, Density function of the proxy material model optimized for the moving deformable component method, k0 is the element stiffness matrix without the elastic modulus, and NE is the number of component mapping elements.

[0019] The sensitivity of the objective function is:

[0020]

[0021] The constraint sensitivity is:

[0022]

[0023] Where V(D) is the total volume, expressed as:

[0024]

[0025] Wherein, V1 is the volume of the first material, V2 is the volume of the second material, f=V2 / V1, DW and DH are the length and width of the design domain, and EW and EH are the length and width of the unit, respectively.

[0026] The weighted average value of the sensitivity of each unit within the filtering radius is used to replace the sensitivity value of the central unit. The unit sensitivity value after the objective function corrects the design variables is:

[0027]

[0028] Among them, H i is the weight operator, and its expression is:

[0029] H i =r min -dist(e,i),{i∈N,|dist(e,i)≤r min},

[0030] Where dist(e,i) is the distance between the centers of units e and i, r min is the filter radius, and N is the number of units.

[0031] The beneficial effects of the present invention are as follows: the method of sequential topology optimization is adopted, the topology structure optimization of the mobile deformable component method is first performed, and then the material selection optimization of the variable density method is performed, and the two are connected by using the proxy material model identification mapping unit, and the internal material layout change does not affect the external topology structure, and does not increase the additional calculation amount. Since the topology structure has been optimized in advance, the design variables of material selection are reduced during material selection optimization, which can significantly improve the calculation efficiency. The method is simple and easy to implement, and also has a wide range of application prospects, especially providing a reference for the local reinforcement structure design of parts and components and the further lightweighting of parts and components. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 These are the optimization results of different Young's moduli based on the MMC method.

[0033] Figure 2 It is a problem model.

[0034] Figure 3 This is a schematic diagram of the MMC method component description.

[0035] Figure 4 It is the optimization result of the short beam problem under different f.

[0036] Figure 5 It is the optimization result of MBB beam problem under different f.

[0037] Figure 6 It is the optimization result of the thin-walled short beam problem under different f. DETAILED DESCRIPTION

[0038] The technical solution of the present invention is described clearly and completely below in conjunction with the accompanying drawings and specific implementation methods. The specific contents listed in the following embodiments are not limited to the technical features necessary for the technical problems to be solved by the technical solutions recorded in the claims. At the same time, the enumerated embodiments are only part of the present invention, not all embodiments.

[0039] The optimization method for multi-material structure design of the present invention adopts a sequential optimization method, first determining the structural topology, and then determining the distribution of materials within the known structural topology. During the optimization process, structural topology optimization and material selection optimization occur sequentially.

[0040] The present invention adopts the moving morphable component method (MMC) for the structural topology of the proxy material model, which defines the design domain of the proxy material model and the components in the design domain. Each component is explicitly described by an explicit topological description function, and the proxy material model is optimized by operations such as component movement, deformation, and coverage and topological optimization. After the optimal topological structure is determined by topological optimization through the moving deformable component method, the variable density method (solid isotropic material with penalization, SIMP) is used to optimize the material selection of each unit in the topological structure to obtain a multi-material structure. The following is a detailed description of only the key points of the present invention, and the general contents of the MMC method and the SIMP method refer to the prior art.

[0041] The optimization process of the method of the present invention is:

[0042] (1) Use the mobile deformable component method to obtain the optimization result of the optimal topological structure and determine the topological description equation φ of all components of the proxy material model under the optimal topological structure S (x);

[0043] (2) According to the topological description equation φ S (x), determine the component mapping unit e=[e1,…,e i ,…,e m ], m is the total number of mapping units, including solid units and boundary units;

[0044] (3) Construct a material selection model, determine the design variables of the variable density method, and map the unit e of each component within the scope of the optimal topology. i Select the materials;

[0045] (4) Perform finite element analysis and sensitivity analysis, update the design variables of the variable density method and check the convergence.

[0046] 1. Component mapping unit identification

[0047] In the present invention, the MMC method and the SIMP method are connected by using a proxy material model to identify a mapping unit. The proxy material model based on the optimal topological structure after optimization by the MMC method identifies the component mapping unit.

[0048] The proxy material model can be expressed as:

[0049]

[0050] Where H is the Heaviside function, i=1,…,4 are the topological function values ​​of the four nodes of element e, and q is the penalty factor (usually q=2). In order to ensure the stability of numerical implementation, H(x) is usually regularized as follows in finite element analysis:

[0051]

[0052] In the formula, τ is a regularization parameter that controls the width of the smooth transition of the Heaviside function, and α is a small positive number that simulates the stiffness of the hole structure in the proxy material model to avoid singular phenomena in the overall stiffness matrix of the structure. Therefore, for the component mapping unit, the unit density should be greater than α q .

[0053] The proxy material model describes the density of the unit based on the topological function value of the node, and divides the unit into hole density unit, intermediate density unit and solid density unit. Taking the four-node bilinear unit as an example, the unit density of the hole density unit is 0, indicating that there is no material. The unit density of the intermediate density unit includes [1 / 4, 2 / 4, 3 / 4], which is the boundary unit of the component boundary. The unit density of the component solid unit is 1. Therefore, the topology optimization method based on the proxy material model only needs to identify the unit with a unit density greater than 0, which is the component mapping unit under the mapping of the topological description equation.

[0054] 2. Unit material selection

[0055] The proxy material model optimized by the MMC method uses one material (i.e., the basic material) for topological structure optimization. The proxy material model is only used to identify whether the unit contains the material, and then the material selection is required for the unit containing the material. The present invention takes two-phase material as an example. The MMC method has determined one material, and the SIMP method only needs to determine whether the mapping unit selects the second material. Therefore, there is only one variable in the SIMP method.

[0056] Taking the optimization of the selection of two-phase materials for each component mapping unit as an example, the following model can be used for unit material selection:

[0057]

[0058] In the formula, ρ e Select the density for the material, E1 and E2 are the Young's modulus of material 1 and material 2 respectively, and p is the penalty coefficient, p ≥ 3. Other multiphase material models can be deduced in the same way.

[0059] 3. Question List

[0060] Since the MMC method is used to obtain initial data (solid material area) and material selection is achieved through the SIMP method on this basis, the SIMP problem formulation and numerical implementation are mainly analyzed.

[0061] Taking the minimum flexibility of the structure as the objective function and the total volume of the structure and the volume ratio of each material relative to the base material as the constraints, the problem is described as follows:

[0062] Find D=(ρ1,ρ2…,ρ NE ) T

[0063]

[0064] Where D is the material selection density variable, and its number is the number of component mapping units NE; C = U T KU is the structural flexibility, u e , k0 are the unit displacement and the unit stiffness matrix without Young's modulus, E e (ρ e ) is the unit density function; F=KU is the finite element analysis control equation; V(D) is the total volume of the structure optimized by the MMC method; V i is the volume of the i-th material, which needs to satisfy m=1,2,3…;V1 is the base material, f i is the volume ratio constraint of the ith material to the base material.

[0065] 4. Finite element analysis

[0066] The key to finite element analysis is to calculate the overall stiffness matrix. Based on the topology optimization method of the present invention, taking two-phase material as an example, the component mapping unit stiffness matrix can be expressed as:

[0067]

[0068] The overall stiffness matrix is:

[0069]

[0070] in, Density function of the proxy material model optimized for the moving deformable component method, k0 is the element stiffness matrix without the elastic modulus, and NE is the number of component mapping elements.

[0071] 5. Sensitivity analysis

[0072] Objective function sensitivity:

[0073]

[0074] From the constraints, we can see that the volume of the first material (the ratio of the volume occupied by the material to the total volume) is:

[0075]

[0076] Where DW and DH are the length and width of the design domain, and EW and EH are the length and width of the unit, respectively.

[0077] The total volume can be expressed as:

[0078]

[0079] Wherein, f=V2 / V1, V1 is the volume of the first material, and V2 is the volume of the second material.

[0080] The constraint sensitivity can be expressed as:

[0081]

[0082] In order to avoid the checkerboard phenomenon and reduce grid dependence, this paper adopts sensitivity filtering technology, that is, the weighted average value of the sensitivity of each unit within the filtering radius is used to replace the sensitivity value of the central unit. The unit sensitivity value after the objective function corrects the design variable is:

[0083]

[0084] In the formula, H i is the weight operator, and its expression is:

[0085] H i =r min -dist(e,i),{i∈N,|dist(e,i)≤r min}

[0086] Where dist(e,i) is the distance between the centers of units e and i, r min is the filter radius, and N is the number of units.

[0087] The updating and convergence test of the design variables of the new variable density method can adopt the existing technology. Taking two materials as an example, the SIMP method can adopt the optimization criterion method to update the design variables without adding additional calculation costs.

[0088] Figure 1 The topological structures obtained under different Young's moduli using the MMC method are shown. It can be seen that the topological structures obtained by setting different Young's moduli are also relatively similar, which also shows the feasibility of the sequential optimization idea. The selection of sequential optimization materials is based on the previous structural topology, that is, the structural topology optimization is not affected by the selection of multiple materials. Its structural topology is determined by a single material. The obtained structure may not be the optimal structure under multiple materials, but the sequential optimization idea is easy to implement and the convergence is controllable, and it also has great application prospects. For example, after obtaining the optimized structure of a certain component, high-strength and lightweight materials can be replaced locally, which can further achieve structural lightweighting and improve structural stiffness.

[0089] The following short beam problem and MBB beam problem are used as analysis objects to verify the feasibility of the proposed method. The problem model is as follows: Figure 2 As shown in the figure, (a) is a short beam problem, (b) is an MBB beam problem, and F is the load. Both numerical examples use minimum flexibility as the objective function and the total volume and volume ratio of the two-phase material as constraints. In order to simplify the analysis, it is assumed that all relevant quantities in the research problem are dimensionless, and the thickness of the design domain is set to a unit value. The Young's modulus is E1=1 and E2=2, respectively, and the Poisson's ratio is v s =0.3, total volume constraint V(D)=0.4, the volume ratios of the two-phase materials are set to f=V2 / V1=[0.2, 0.5, 0.8, 1, 1.5, 2], penalty index p=3, load F=1, and penalty index q=2.

[0090] The components in the MMC method adopt secondary variable width components, such as Figure 3 As shown, the components are described using the hyperelliptic equation:

[0091]

[0092] in,

[0093]

[0094] Where φ(x, y) represents the topological function value of the coordinates corresponding to the entire design domain D, the parameter p is a large positive even number, and p = 6 is generally taken according to numerical experience, L is the half length of the component, (x′, y′) and (x, y) represent the coordinate position of any point in the local coordinate system and the global coordinate system respectively, θ is the rotation angle of the component (the angle from the global coordinate system to the local coordinate system), and f(x′) is a function that describes the shape of the component. For a quadratic variable width component, it can be expressed as:

[0095]

[0096] Both the short beam problem and the MMB beam problem adopt Figure 3 The components shown in the figure have a design domain size of 2×1 for the short beam problem and a grid resolution of 80×40, and a design domain size of 3×1 for the MBB beam problem and a grid resolution of 90×30. In addition, a single hole thin-walled structure component (scaling factor a=b=0.5) is used to optimize the short beam problem and obtain a thin-walled short beam structure. The problem model is still the same as Figure 2 As shown in (a), the design domain size is 2×1 and the grid resolution is set to 120×60. The optimization results under different f are shown in Figure 4 , Figure 5 , Figure 6As shown in the figure, in the final optimized structure, the black part within the boundary is the E1 material distribution, and the blank area is the E2 material distribution. It can be seen that the proposed method can realize the secondary material layout of the MMC method optimization result to further reduce the structural flexibility, and as f increases, the structural flexibility decreases. Since the secondary material layout is realized based on the density method, it can not only generate a more flexible material layout by utilizing the advantages of the high design freedom of the density method, but also has better adaptability to complex structures. The proposed method has clear and explicit external topological structure boundaries, clear internal multi-material layout, and fewer numerical problems such as checkerboard and grayscale units.

[0097] The optimization results show the effectiveness of the MMC-SIMP sequential topology optimization method of the present invention, and multi-material structure design can be realized according to the optimization results. Since the overall effective structure has been determined by the MMC method, the adjustability of the local reinforcement structure obtained by the SIMP method is relatively strong, which is convenient for post-processing some discontinuous structures, relatively isolated structures and difficult-to-process structures, and will not fundamentally affect the overall structural performance.

[0098] In this method, a sequential optimization method is adopted to determine the structural topology first and then the material layout, wherein the structural topology is determined by the MMC method, the internal material layout of the structure is determined by the SIMP method, and the SIMP method optimizes the material selection and only constructs the mapping unit material selection stiffness matrix, thereby reducing the design variables of the material selection. The method of the present invention can make full use of the advantages of the two topology optimization methods, and the data crossover between different optimization methods is small, so it is simple and easy to implement and has wide applicability. This sequential topology optimization method not only has an explicit optimized structural expression, but also can fully express the design details, and the design variables are also greatly reduced compared to the single SIMP method. Through the numerical example analysis of two-phase materials at different material volume ratios, it is shown that the proposed method can obtain an effective multi-material structure, and the structural boundary and internal material layout are clear, which confirms the feasibility of the proposed method. Although the method of topological structure and material parallel optimization may be slightly worse than the structural optimization accuracy, this method is applicable to the working condition where the topological structure has been determined, but the overall strength needs to be improved by using multiple materials, and has broad application prospects, especially for the local reinforcement structure design of parts and components and the further lightweight of parts and components. Provide reference, such as adding surface coatings, local fiber reinforcement, etc.

[0099] The above description of the specific implementation mode is only used to help understand the technical concept and core idea of ​​the present invention. Although the technical solution is described and illustrated using a specific preferred embodiment, it should not be understood as a limitation of the present invention itself. Those skilled in the art may make various changes in form and details without departing from the technical concept of the present invention. These easily conceived changes or substitutions should all be included in the protection scope of the present invention.

Claims

1. A sequential topology optimization method for multi-material structure design, characterized in that: firstly, a proxy material model design domain and components are defined, a moving deformable component method is used to perform topology optimization to determine the optimal topology structure, and then a variable density method is used to optimize the material selection of each unit in the topology structure to obtain a multi-material structure, comprising the following steps: (1) Use the mobile deformable component method to obtain the optimization result of the optimal topological structure and determine the topological description equations of all components of the proxy material model under the optimal topological structure; (2) According to the topological description equation, determine the component mapping units under the mapping of the topological description equation, including entity units and boundary units; (3) constructing a material selection model, determining the design variables of the variable density method, and selecting the material of each component mapping unit within the range of the optimal topological structure; (4) Perform finite element analysis and sensitivity analysis, update the design variables of the variable density method and check the convergence.

2. A sequential topology optimization method for multi-material structure design as described in claim 1, characterized in that: after obtaining the optimal topological structure of the proxy material model by moving the deformable component method, the topological function value of each unit of the proxy material model is determined based on the topological description equation, and the unit with a unit density greater than 0 is identified and determined as the component mapping unit under the mapping of the topological description equation.

3. A sequential topology optimization method for multi-material structure design as described in claim 1, characterized in that: a moving deformable component method is used to perform topology optimization based on one material and determine the optimal topology structure, and then a variable density method is used to determine whether the component mapping unit selects a second material to achieve material selection optimization.

4. A sequential topology optimization method for multi-material structure design as claimed in claim 3, characterized in that: a variable density method is used to optimize the selection of two-phase materials for each component mapping unit, and a material selection model is constructed as follows: where ρ e Select density for the material, E1 and E2 are the Young's modulus of the two materials respectively, and p is the penalty coefficient, p ≥ 3.

5. A sequential topology optimization method for multi-material structure design as described in claim 4, characterized in that: the variable density method takes the minimum flexibility of the structure as the objective function, and takes the total volume of the structure and the volume proportion of each material relative to the base material as constraints to optimize the material density selection variables.

6. A sequential topology optimization method for multi-material structure design as claimed in claim 5, characterized in that: a material selection stiffness matrix of a component mapping unit is constructed for finite element analysis, and the component mapping unit stiffness matrix can be expressed as: The overall stiffness matrix is in, Density function of the proxy material model optimized for the moving deformable component method, k0 is the element stiffness matrix without the elastic modulus, and NE is the number of component mapping elements.

7. A sequential topology optimization method for multi-material structure design as claimed in claim 6, characterized in that: the sensitivity of the objective function is: The constraint sensitivity is: in, V(D) is the total volume, expressed as: Wherein, V1 is the volume of the first material, V2 is the volume of the second material, f=V2 / V1, DW and DH are the length and width of the design domain, and EW and EH are the length and width of the unit, respectively.

8. A sequential topology optimization method for multi-material structure design as claimed in claim 7, characterized in that: the weighted average value of the sensitivity of each unit within the filtering radius is used to replace the sensitivity value of the central unit, and the unit sensitivity value after the objective function corrects the design variable is: in, H i is the weight operator, and its expression is: H i =r min -dist(e,i),{i∈N,|dist(e,i)≤r min }, Where dist(e,i) is the distance between the centers of units e and i, r min is the filter radius, and N is the number of units.

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