Topological optimization method for composite material with stretching and compression characteristics

Abstract

The invention provides a topological optimization method for a tension / compression structure. The method is based on a three-phase material model formed by pure tension, pure compression and empty attribute materials. According to the method, the discrete design domain of the mixed stress unit is adopted, the unit tension / pressure state judgment criterion based on the principal stress value and the tension / pressure tolerance is formulated, the tension / pressure state of the unit can be more accurately determined, and therefore material distribution based on the unit stress state is achieved. And finally, deriving the sensitivity of the target function to the design variable, and filtering the sensitivity by using a three-field filtering technology.

Description

[Technical field]

[0001] The invention relates to the technical field of composite material engineering structure design, and in particular to a composite material topology optimization method with tensile and compressive properties. [Background technology]

[0002] Currently, most of the research in the field of topology optimization focuses on the optimization of a single material. However, the transition from focusing on single-material topology optimization technology to multiphase material topology optimization is an inevitable trend in the development of science and technology. This transition stems from the fact that multiphase material structures can meet new design concepts, including lightweight structures, diversified designs, and functionalized products. At the same time, multiphase materials themselves have important characteristics such as excellent mechanical properties, strong corrosion resistance, and long service life.

[0003] Many materials in actual engineering exhibit completely different mechanical behaviors under tension and compression, that is, they have tension / compression asymmetric characteristics, such as reinforced concrete materials, fiber-reinforced composite materials, etc. Such materials are called dual-modulus materials, which have different elastic moduli under tension and compression. This method takes into account the tension / compression asymmetric characteristics of the material, arranges materials suitable for tension in the tension area of ​​the structure, and arranges materials suitable for compression in the compression area, simulating the optimal distribution of dual-modulus materials. By utilizing the different mechanical properties of dual-modulus materials under tension and compression, better design results can be achieved under different load conditions. [Summary of the invention]

[0004] In view of the above, the present invention provides a composite material topology optimization method with tensile and compressive characteristics, which distributes the tensile / compressive composite material on the structure based on the unit stress sign and tolerance limit. The material distribution accuracy is improved by iterative calculation and sensitivity analysis. This method at least solves the limitation problem of inaccurate material distribution in the topology optimization of composite material structures in the prior art.

[0005] To achieve the above object, the technical solution of the present invention is:

[0006] A method for topological optimization of composite materials with tensile and compressive properties comprises the following steps:

[0007] S1. A topology optimization model is established under volume constraints to minimize the structural flexibility; the constraints include equilibrium equations;

[0008] S2. Discretization of design domain: The design domain to be optimized for the composite structure is discretized into N units, and an initial density variable x is assigned to each unit. e , and assuming that the initial design domain is all tensile materials, obtain the initial elastic modulus matrix

[0009] S3. Finite element iterative solution of the topology optimization model of composite structures with tensile and compressive characteristics:

[0010] S31. Calculate the initial element stiffness matrix and the element principal stress σ 1 , σ 2 ;

[0011] S32. Judgment criteria, based on the unit stress σ e Determine the tensile / compressive characteristics of the unit, and assign the corresponding material tensile / compressive properties and corresponding elastic modulus to each unit;

[0012] S33. According to the new elastic modulus matrix E e Calculate the stiffness matrix k e , and then assemble the global stiffness matrix K(σ)

[0013] S34. Perform finite element solution according to the equilibrium equation K(σ)U=F to obtain the displacement matrix U;

[0014] S35. Loop S31-S34, and enter S4 when the relative change of the Euclidean norm of the structural displacement matrix in two adjacent iteration steps is less than the preset tolerance limit tol; otherwise, return to S3;

[0015] S4. Based on the chain rule, the derivative of the objective function with respect to the design variables is obtained;

[0016] S5. Optimize the cell density variable x based on the moving asymptote method (MMA) e ;

[0017] S6. Repeat S3-S5 until the objective function converges.

[0018] Preferably, the topology optimization model of the composite material with tensile and compressive properties is specifically:

[0019] min c(x)=U T K(σ)U

[0020] stK(σ)U=F

[0021]

[0022] 0≤x e ≤1

[0023] Where c is the structural flexibility, K, U and F are the overall stiffness matrix, displacement vector and load vector respectively, V max is the volume of the structure when it is filled with material, v fis a given volume fraction, and N is the total number of finite element units. Since the two materials with different properties in the tension / compression structure topology optimization have different elastic moduli, the overall stiffness K(σ) of the structure is related to the stress of each unit, which can be obtained by assembling the stiffness matrix of the tension area and the stiffness matrix of the compression area.

[0024] Preferably, in S31, the initial unit stiffness matrix is ​​calculated and the element principal stress σ 1 , σ 2 Specifically:

[0025]

[0026] In the formula, x min It is the minimum value of the unit density given to prevent the stiffness matrix from being singular, corresponding to the empty attribute material, p is the penalty factor, E e (σ e ) is the overall elastic modulus matrix, σ e Indicates that the elastic modulus is related to the unit stress, B is the strain-displacement matrix, u e is the unit displacement matrix, x min The minimum value of the element density is assigned to prevent the stiffness matrix from being singular, corresponding to the empty attribute material, x min =10 -6 .

[0027] Preferably, in S32, according to the judgment criteria, according to the unit stress σ e The tension / compression characteristics of the judgment unit are specifically:

[0028] When σ 1 and σ 2 If σ has the same sign, 1 > 0, the unit is given tensile material properties, and E e =E T ; If σ 1 <0, the unit is given compressive material properties, and E e =E C ;

[0029] When σ 1 and σ 2 When the signs are opposite, if σ 1 +σ 2 > 0, the unit is given tensile material properties, and E e =E T ; If σ 1 +σ 2 <0, the unit is given compressive material properties, and E e =E C ;

[0030] When σ1 and σ 2 When the absolute values ​​of are all less than the predetermined tolerance limit tol, the element is assigned the empty material property, E e =E min .

[0031] Preferably, in S33, according to the new elastic modulus matrix E e The calculation stiffness matrix is ​​specifically expressed as:

[0032]

[0033] In the formula, E e (σ e ) indicates that the elastic modulus of the element is related to the stress of the element, and B is the strain-displacement matrix of the element.

[0034] Preferably, in S4, according to the chain derivation rule, the objective function c is obtained for the design variable x e The specific contents of the derivative include:

[0035] Objective function J for design variable ρ e The derivative of is:

[0036]

[0037] Preferably, in S5, the unit density variable is optimized based on the moving asymptote method (MMA) as follows:

[0038] The neighborhood set N of the e-th mixed stress unit e The definition is as follows:

[0039] N e ={q|||y q -y e ||≤r min}

[0040] Among them, y q represents the centroid position of the qth unit, y e is the centroid position of the e-th unit, r min represents the filter radius. Then the density variable of the e-th unit is The expression is:

[0041]

[0042] in, represents the volume of the qth unit when it is full of material, w e,q (y q -y e ) represents the weight value of the qth unit in constructing the relative density variable of the eth unit;

[0043] Introducing the nonlinear projection expression, the density variable Replace with physical variables The expression can be expressed as:

[0044]

[0045] Where β is the set projection parameter and η is the set threshold.

[0046] Through the above technical solution, compared with the prior art, the present invention discloses a method for topological optimization of composite materials with tensile and compressive properties, which has the following benefits:

[0047] First, based on the tension / compression structure modeling method of composite materials, the tension / compression structure is divided into tensile, compressive and empty material areas. According to the asymmetric tension / compression characteristics of the material, the material with better tensile performance is distributed in the tensile area of ​​the structure, while the material with better compression performance is distributed in the compression area of ​​the structure, which improves the utilization efficiency of the material. A new tension / compression material judgment method is constructed, and an efficient sensitivity calculation formula is proposed.

Brief Description of the Drawings

[0048] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0049] Figure 1 It is a schematic flow chart of a topology optimization method for a composite material with tensile and compressive properties according to the present invention;

[0050] Figure 2 A schematic diagram of a bridge structure provided by an embodiment of the present invention;

[0051] Figure 3 Optimize design results for bridge studies; [Specific implementation method]

[0052] S1. A topology optimization model is established under volume constraints to minimize the structural flexibility; the constraints include equilibrium equations;

[0053] S2. Discretization of design domain: The design domain to be optimized for the composite structure is discretized into N units, and an initial density variable x is assigned to each unit. e , and assuming that the initial design domain is all tensile materials, obtain the initial elastic modulus matrix

[0054] S3. Finite element iterative solution of the topology optimization model of composite structures with tensile and compressive characteristics:

[0055] S31. Calculate the initial element stiffness matrix and the element principal stress σ 1 , σ 2 ;

[0056] S32. Judgment criteria, based on the unit stress σ e Determine the tensile / compressive characteristics of the unit, and assign the corresponding material tensile / compressive properties and corresponding elastic modulus to each unit;

[0057] S33. According to the new elastic modulus matrix E e Calculate the stiffness matrix k e , and then assemble the global stiffness matrix K(σ)

[0058] S34. Perform finite element solution according to the equilibrium equation K(σ)U=F to obtain the displacement matrix U;

[0059] S35. Loop S31-S34, and enter S4 when the relative change of the Euclidean norm of the structural displacement matrix in two adjacent iteration steps is less than the preset tolerance limit tol; otherwise, return to S3;

[0060] S4. Based on the chain rule, the derivative of the objective function with respect to the design variables is obtained;

[0061] S5. Optimize the cell density variable x based on the moving asymptote method (MMA) e ;

[0062] S6. Repeat S3-S5 until the objective function converges.

[0063] In order to further implement the above technical solution, the topology optimization model of composite materials with tensile and compressive properties is optimized as follows:

[0064] min c(x)=U T K(σ)U

[0065] stK(σ)U=F

[0066]

[0067] 0≤x e ≤1

[0068] Where c is the structural flexibility, K, U and F are the overall stiffness matrix, displacement vector and load vector respectively, V max is the volume of the structure when it is filled with material, v fis a given volume fraction, and N is the total number of finite element units. Since the two materials with different properties in the tension / compression structure topology optimization have different elastic moduli, the overall stiffness K(σ) of the structure is related to the stress of each unit, which can be obtained by assembling the stiffness matrix of the tension area and the stiffness matrix of the compression area.

[0069] In order to further implement the above scheme, in S31, the initial unit stiffness matrix is ​​calculated and the element principal stress σ 1 , σ 2 Specifically:

[0070]

[0071] In the formula, x min It is the minimum value of the unit density given to prevent the stiffness matrix from being singular, corresponding to the empty attribute material, p is the penalty factor, E e (σ e ) is the overall elastic modulus matrix, σ e Indicates that the elastic modulus is related to the unit stress. B is the strain-displacement matrix, u e is the unit displacement matrix.

[0072] In order to further implement the above scheme, in S32, according to the judgment criteria, according to the unit stress σ e The tension / compression characteristics of the judgment unit are specifically:

[0073] When σ 1 and σ 2 If σ has the same sign, 1 > 0, the unit is given tensile material properties, and E e =E T ; If σ 1 <0, the unit is given compressive material properties, and E e =E C ;

[0074] When σ 1 and σ 2 When the signs are opposite, if σ 1 +σ 2 > 0, the unit is given tensile material properties, and E e =E T ; If σ 1 +σ 2 <0, the unit is given compressive material properties, and E e =E C ;

[0075] When σ 1 and σ 2 When the absolute values ​​of are all less than the predetermined tolerance limit tol, the element is assigned the empty material property, Ee =E min .

[0076] In order to further implement the above scheme, in S33, according to the new elastic modulus matrix E e The calculation stiffness matrix is ​​specifically expressed as:

[0077]

[0078] In the formula, E e (σ e ) indicates that the elastic modulus of the element is related to the stress of the element, and B is the strain-displacement matrix of the element.

[0079] In order to further implement the above scheme, in S4, according to the chain derivation rule, the objective function c is obtained for the design variable x e The specific contents of the derivative include:

[0080] Objective function J for design variable ρ e The derivative of is:

[0081]

[0082] In order to further implement the above scheme, in S5, the unit density variable is optimized based on the moving asymptote method (MMA) as follows:

[0083] The neighborhood set N of the e-th mixed stress unit e The definition is as follows:

[0084] N e ={q|||y q -y e ||≤r min}

[0085] Among them, y q represents the centroid position of the qth unit, y e is the centroid position of the e-th unit, r min represents the filter radius. Then the density variable of the e-th unit is The expression is:

[0086]

[0087] in, represents the volume of the qth unit when it is full of material, w e,q (y q -y e ) represents the weight value of the qth unit in constructing the relative density variable of the eth unit;

[0088] Introducing the nonlinear projection expression, the density variable Replace with physical variables The expression can be expressed as:

[0089]

[0090] Where β is the set projection parameter and η is the set threshold.

[0091] The following example will verify the effectiveness of the proposed method. The elements under tension and compression are represented by red and blue, respectively, that is, red represents the tensile material and blue represents the compressive material, where the elastic modulus of the compressive material is a fixed value E c =200GPa, define R=E t / E c , changing the value of R can obtain two materials with different elastic moduli, the penalty factor p = 3, and the maximum number of iterations is set to 200.

[0092] like Figure 1 The first convergence criterion is that the relative change of the Euclidean norm of the structural displacement matrix in two adjacent iterations is less than the preset tolerance limit; the second convergence criterion is that the sum of the objective functions of the last two iterations is less than the preset tolerance limit.

[0093] like Figure 2 In the bridge example shown, a 60m×71m rectangular bridge is considered as the design domain for optimization, which is discretized into 68160 (240×284) units. The four vertices of the design domain are completely fixed, and the bridge deck with a thickness of 1m in the middle is regarded as a non-design domain. A deterministic load P=500N / m is applied to the bridge deck in a vertical downward direction. Other parameters are set as: volume fraction v f =0.2, the filter radius is r min = 3. Three sets of optimization designs with different tensile / compressive modulus ratios were set: (a) R = 1 (i.e., the tensile and compressive moduli are the same), (b) R = 2 (i.e., the tensile performance of the material is better), and (c) R = 2 / 3 (i.e., the compressive performance of the material is better).

[0094] The design results are as follows Figure 3 shown. Figure 3 In the figure, the upper part of the design domain is mainly in tension, while the lower part is in compression. Figure 3 In (a), since the tensile material and the compressive material have the same modulus, the design result includes both the tensile material and the compressive material of equal volume. It is worth noting that even if the tensile modulus and the compressive modulus of the material are the same, the design result is still not vertically symmetrical. This is because the load is applied to the bridge deck, and the existence of the non-design domain makes the designable part of the structure no longer symmetrical about the horizontal direction.

[0095] The present invention verifies the effectiveness of the proposed method through the above examples. The results of the examples show that there are obvious differences between the optimization results under the condition of load uncertainty and the condition of load certainty, and the optimization results obtained under the condition of load uncertainty have better mechanical properties and are more in line with engineering practice. The topological optimization method of tension / compression composite materials under load uncertainty designed by the present invention can effectively and accurately distribute two materials with different tension / compression properties under the condition of load uncertainty, and in all design results, the distribution of materials has clear boundaries, and the obtained design results have better mechanical properties. If the load is applied to the bottom surface of the non-design domain and the direction is upward, the obtained design result will be different from Figure 3 (a) Symmetric (e.g. Figure 3 (b)). When the modulus of the tensile material is greater than that of the compressive material, the optimal design result only exists in the upper part of the design domain, as shown in Figure 3 (c) shows that, similarly, if the modulus of the tensile material is smaller than that of the compressive material, the optimal design result only exists in the lower part of the design domain, such as Figure 3 (d). This is the result of the optimization of the tension / compression structure model to adapt to different modulus ratios. Changing the elastic modulus ratio of the two materials can obtain different design results and get optimization results close to pure tension or pure compression. In addition, it can be seen from the structural flexibility of the optimization results that due to Figure 3 The two design results (a) and (b) only change the direction of the load action, without changing the elastic modulus ratio of the material, so the two results have similar objective function values. Figure 3 (c) and (d), it is found that when R = 2, the objective function value of the structure with only distributed tensile material is lower. This is because the distributed tensile material has a larger elastic modulus. This structure has the best mechanical properties among the three groups of structures with different tensile / compressive elastic moduli.

[0096] The above embodiments are only used to illustrate the technical solutions of the present application, rather than to limit them. Although the present application has been described in detail with reference to the aforementioned embodiments, a person skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features may be exchanged equivalently. Such modifications or substitutions do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present application, and should all be included in the protection scope of the present application.

Claims

1. A method for topological optimization of composite materials with tensile and compressive properties, characterized in that: The optimization method comprises the following steps: S1. A topology optimization model is established under volume constraints to minimize the structural flexibility; the constraints include equilibrium equations; S2. Discretization of design domain: The design domain to be optimized for the composite structure is discretized into N units, and an initial density variable x is assigned to each unit. e , and assuming that the initial design domain is all tensile materials, obtain the initial elastic modulus matrix S3. Finite element iterative solution of the topology optimization model of composite structures with tensile and compressive characteristics: S31. Calculate the initial element stiffness matrix and unit principal stresses σ1, σ2; S32. Judgment criteria, based on the unit stress σ e Determine the tensile / compressive characteristics of the unit, and assign the corresponding material tensile / compressive properties and corresponding elastic modulus to each unit; S33. According to the new elastic modulus matrix E e Calculate the stiffness matrix k e , and then assemble the global stiffness matrix K(σ) S34. Perform finite element solution according to the equilibrium equation K(σ)U=F to obtain the displacement matrix U; S35. Loop S31-S34, and enter S4 when the relative change of the Euclidean norm of the structural displacement matrix in two adjacent iteration steps is less than the preset tolerance limit tol; otherwise, return to S3; S4. Based on the chain rule, the derivative of the objective function with respect to the design variables is obtained; S5. Optimize the cell density variable x based on the moving asymptote method (MMA) e ; S6. Repeat S3-S5 until the objective function converges.

2. The method for topological optimization of composite materials with tensile and compressive properties according to claim 1, characterized in that: In S1, the topology optimization model of composite materials with tension and compression characteristics is as follows: where c(x)=U T K(σ)U stK(σ)U=F 0≤x e ≤1 Where c is the structural flexibility, K, U and F are the overall stiffness matrix, displacement vector and load vector respectively, V max is the volume of the structure when it is filled with material, v f is a given volume fraction, N is the total number of finite element units; since the two materials with different properties in the tension / compression structure topology optimization have different elastic moduli, the overall stiffness K(σ) of the structure is related to the stress of each unit, which can be assembled from the tensile region stiffness matrix and the compression region stiffness matrix.

3. The method for topological optimization of composite materials with tensile and compressive properties according to claim 1, characterized in that: In S31, calculate the initial element stiffness matrix And the unit principal stresses σ1 and σ2 are specifically: In the formula, x min It is the minimum value of the unit density given to prevent the stiffness matrix from being singular, corresponding to the empty attribute material, p is the penalty factor, E e (σ e ) is the overall elastic modulus matrix, σ e Indicates that the elastic modulus is related to the unit stress, B is the strain-displacement matrix, u e is the unit displacement matrix.

4. The method for topological optimization of composite materials with tensile and compressive properties according to claim 1, characterized in that: In S32, according to the judgment criteria, based on the unit stress σ e The tension / compression characteristics of the judgment unit are specifically: When σ1 and σ2 have the same sign, if σ1>0, the element is assigned tensile material properties, and E e =E T ; If σ1<0, the unit is given compressive material properties, and E e =E C ; When σ1 and σ2 have opposite signs, if σ1+σ2>0, the element is given tensile material properties, and E e =E T ; If σ1+σ2<0, the element is given compressive material properties, and E e =E C ; When the absolute values ​​of σ1 and σ2 are both less than the predetermined tolerance limit tol, the element is assigned the empty material property, E e =E min。 5. The method for topological optimization of composite materials with tensile and compressive properties according to claim 1, characterized in that: In S33, according to the new elastic modulus matrix E e The calculation stiffness matrix is ​​specifically expressed as: In the formula, E e (σ e ) indicates that the elastic modulus matrix of the unit is related to the stress of the unit, and B is the strain-displacement matrix of the unit.

6. The topology optimization method of composite materials with tensile and compressive properties according to claim 1, characterized in that: In S4, according to the chain rule, we can get the objective function c for the design variable x e The specific contents of the derivative include: Objective function J for design variable ρ e The derivative of is:

7. The method for topological optimization of composite materials with tensile and compressive properties according to claim 1, characterized in that: In S5, the cell density variables are optimized based on the moving asymptote method (MMA) as follows: The neighborhood set N of the e-th mixed stress unit e The definition is as follows: N e ={q|||y q -and e ||≤r min } Among them, y q represents the centroid position of the qth unit, y e is the centroid position of the e-th unit, r min represents the filter radius, then the density variable of the e-th unit The expression is: in, represents the volume of the qth unit when it is full of material, w e,q (y q -y e ) represents the weight value of the qth unit in constructing the relative density variable of the eth unit; Introducing the nonlinear projection expression, the density variable Replace with physical variables The expression can be expressed as: Where β is the set projection parameter and η is the set threshold.

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