A three-dimensional fiber-reinforced composite material multi-scale variable stiffness optimization design method

Abstract

The application discloses a three-dimensional fiber-reinforced composite material multi-scale variable stiffness optimization design method, relates to the field of fiber-reinforced composite material variable stiffness design, and establishes a three-dimensional fiber-reinforced composite material multi-scale variable stiffness optimization design framework, and deduces the analytical sensitivity of macrostructure topological design variables and micro-fiber laying angle design variables to an objective function and a volume constraint. Through a numerical example, the effectiveness of the method proposed in the application is verified, the method is proposed based on original calculation data, the highly adaptive design result and the manufacturing result are maximally ensured, and the method has the advantages of strong adaptability and easy popularization.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of fiber-reinforced composite material variable stiffness design, and particularly relates to a three-dimensional fiber-reinforced composite material multi-scale variable stiffness optimization design method. BACKGROUND

[0002] With the rapid development of aerospace technology, the requirements of aircraft structure design for lightweight, high performance and functional integration are becoming increasingly stringent. Under this background, fiber-reinforced composite materials have been widely used due to their high specific strength, large specific modulus and strong designability. Currently, the mainstream design method of this kind of material, whether the fiber laying angle is taken as a direct design variable or the fiber path is described by a hypothetical continuous function or an optimization strategy based on discrete materials, is still limited to the two-dimensional design paradigm due to the difficulty in intuitively and accurately representing the fiber laying angle in three-dimensional space. This traditional two-dimensional design method has inherent limitations when faced with aerospace components with complex configurations. It cannot reflect the distribution of fibers in three-dimensional space and is difficult to carry out multi-scale variable stiffness optimization design of three-dimensional composite materials, resulting in design deviation or performance loss. Therefore, developing a three-dimensional composite material variable stiffness optimization design method that fully considers the spatial distribution of fibers and is suitable for complex configurations is of great engineering significance and urgency to break through the existing technical bottlenecks, fully release the application potential of composite materials, and realize the integrated design and performance leap of aerospace main load-bearing structures.

[0003] However, there are the following difficulties in carrying out multi-scale variable stiffness optimization design of three-dimensional composite materials: 1) difficulty in explicit representation of fiber laying angle in three-dimensional space; 2) how to represent the diversity of fiber laying connection in three-dimensional space; 3) linear increase of design variables with increasing grid density and number of alternative fiber laying angles, bringing the challenge of time-consuming calculation; 4) difficulty in deriving analytical sensitivity for three-dimensional composite material multi-scale variable stiffness optimization design under two-dimensional normal distribution interpolation format. SUMMARY

[0004] In view of the above deficiencies in the prior art, the three-dimensional fiber-reinforced composite material multi-scale variable stiffness optimization design method provided by the present application solves the problems of difficulty in representation, time-consuming calculation and difficulty in deriving analytical sensitivity in the prior art.

[0005] In order to achieve the above-mentioned application purposes, the technical scheme adopted by the present application is as follows: a three-dimensional fiber-reinforced composite material multi-scale variable stiffness optimization design method, comprising the following steps:

[0006] S1: carrying out finite element mesh division on the structure to be designed, obtaining a design unit;

[0007] ​S2: Select the number of alternative rotation angle 1 and alternative rotation angle 2;

[0008] S3: Initialize macroscopic structural topology optimization and microscopic fiber laying angle design variables;

[0009] S4: Calculate the micro-elastic constitutive matrix based on the two-dimensional normal distribution fiber optimization interpolation scheme and the micro-fiber laying angle design variable;

[0010] S5: Calculate the unit elastic constitutive matrix based on the micro-elastic constitutive matrix and macro-structural topology optimization design variables;

[0011] S6: Calculate the element stiffness based on the element elastic constitutive matrix, and assemble the overall stiffness based on the element stiffness.

[0012] S7: Based on the overall stiffness, perform static equilibrium analysis of the structure and obtain the objective function;

[0013] S8: Calculate the volume constraints of the design region;

[0014] S9: Perform volume constraint and objective function sensitivity analysis;

[0015] S10: Enter the optimization iteration loop, repeat steps S4-S9 until the convergence criterion is met, and realize the multi-scale variable stiffness optimization design of three-dimensional fiber reinforced composite materials.

[0016] Furthermore, the formula for calculating the micro-elastic constitutive matrix in S4 is as follows:

[0017] ;

[0018] ;

[0019] ;

[0020] in, For the first The microelastic constitutive matrix of each unit, and These represent the number of alternative rotation angle 1 and alternative rotation angle 2, respectively. For the first The weighting coefficients of the elastic constitutive matrix of the candidate fiber layup angles for each unit. For the normalized first The weighting coefficients of the elastic constitutive matrix of the candidate fiber layup angles for each unit. For the first species, first Elastic constitutive matrix of alternative fiber layup angles. For angle Optimize the penalty coefficient. For angle Optimize the penalty coefficient. For the first The angle of each unit Design variables, the range of which is , For the first The angle of each unit Design variables, the range of which is .

[0021] Furthermore, the formula for calculating the elastic constitutive matrix of the element in S5 is as follows:

[0022] ;

[0023] in, For the first The element elastic constitutive matrix of each element. The design variables for macroscopic structural topology optimization are within the following range. , Design a penalty coefficient for macroscopic topology optimization.

[0024] Furthermore, the element stiffness calculation in S6 is as follows:

[0025] ;

[0026] in, For the first The element stiffness matrix of each element. This is the strain-displacement matrix, with superscripts. To represent the transpose of a matrix, This is a unit domain.

[0027] Furthermore, S7 emphasizes structural flexibility. Minimize to the objective function, as shown in the formula:

[0028] ;

[0029] in, Let be the overall displacement vector of the structure. This is the overall stiffness matrix of the structure.

[0030] Furthermore, the volume constraint calculation formula in S8 is as follows:

[0031] ;

[0032] in, The actual volume of the structure. For the first The unit volume of each cell. This represents the total number of units.

[0033] Further, the volume constraint and the sensitivity analysis of the objective function in S9 are performed, including:

[0034] Macro-structure topology optimization design variable The sensitivity of the objective function is:

[0035] ;

[0036] Structure total stiffness matrix The partial derivative of the design variable is calculated by calculating the element stiffness matrix :

[0037] ;

[0038] ;

[0039] Wherein, is the area to be designed;

[0040] Micro-fiber laying angle design variable , The sensitivity of the objective function , only needs to calculate , :

[0041] The calculation is:

[0042] ;

[0043] ;

[0044] ;

[0045] The calculation is:

[0046] ;

[0047] ;

[0048] ;

[0049] Macro-structure topology optimization design variable , micro-fiber laying angle design variable , The sensitivity of the volume constraint , , is: ​

[0050] , , .

[0051] Further, the convergence criteria in S10 is:

[0052] ;

[0053] wherein, is the convergence criterion, is the convergence tolerance, is a positive integer, representing the continuous iteration steps, is the current iteration step, is the first continuous iteration step.

[0054] The beneficial effects of the present application are:

[0055] (1) The design variables of the method proposed in the present application do not increase linearly with the increase of the alternative rotation angle;

[0056] (2) The method proposed in the present application has the characteristics of non-convex objective function, which can avoid the optimization result falling into local optimum;

[0057] (3) The present application clearly and explicitly gives the derivation of the analytical sensitivity of the volume constraint and the objective function;

[0058] (4) The present application is suitable for any three-dimensional structure of fiber reinforced composite materials, which is different from the current discrete fiber optimization theory system which is only suitable for two-dimensional planar structure. BRIEF DESCRIPTION OF DRAWINGS

[0059] Figure 1 is a flow chart of a three-dimensional fiber reinforced composite material multi-scale variable stiffness optimization design method.

[0060] Figure 2 is a schematic diagram of a three-dimensional cantilever beam calculation structure.

[0061] Figure 3 is a schematic diagram of the three-dimensional fiber reinforced composite material multi-scale variable stiffness optimization design result based on the two-dimensional normal distribution of the present application. DETAILED DESCRIPTION

[0062] The present application will be further described below in conjunction with the drawings and specific embodiments.

[0063] As shown in Figure 1 , a three-dimensional fiber reinforced composite material multi-scale variable stiffness optimization design method comprises the following steps:

[0064] S1: performing finite element meshing on the structure to be designed, to obtain a design unit;

[0065] S2: selecting the number of candidate rotation angle 1 and candidate rotation angle 2;

[0066] The number of candidate rotation angle 1 and candidate rotation angle 2 is respectively , , the range of which is: the range of candidate rotation angle 1 is , and the range of candidate rotation angle 1 is .

[0067] S3: initializing macroscopic structure topology optimization and micro-fiber laying angle design variable;

[0068] Initializing macroscopic structure topology optimization , micro-fiber laying angle design variable , , the number of which is respectively: units , units , units , a total of units; the range of which is respectively: , , ;

[0069] The elastic constitutive matrix of the first , the second candidate fiber laying angle can be expressed as:

[0070]

[0071] The weight coefficient of the elastic constitutive matrix of the first candidate fiber laying angle of the unit can be expressed as:

[0072]

[0073] S4: based on the two-dimensional normal distribution fiber optimization interpolation format and the micro-fiber laying angle design variable, calculating the micro-elastic constitutive matrix;

[0074] The micro-elastic constitutive matrix calculation formula is:

[0075] ;

[0076] ;

[0077] ; ​

[0078] wherein, is the micro elastic constitutive matrix of the th unit, and are the number of alternative rotation angle 1, alternative rotation angle 2 respectively, is the weight coefficient of the elastic constitutive matrix of the alternative fiber laying angle of the th unit, is the normalized weight coefficient of the elastic constitutive matrix of the alternative fiber laying angle of the th unit, is the elastic constitutive matrix of the alternative fiber laying angle of the th, the th alternative fiber laying angle, is the angle optimization penalty coefficient, is the angle optimization penalty coefficient, is the angle design variable of the th unit, the range of which is , is the angle design variable of the th unit, the range of which is .

[0079] The weight coefficient is normalized to meet the physical properties of the material, and the weight coefficient of the elastic constitutive matrix of the alternative fiber laying angle of the th unit is:

[0080]

[0081] S5: Calculate the unit elastic constitutive matrix based on the micro elastic constitutive matrix and the macro structure topology optimization design variable;

[0082] Considering the concurrent multi-scale variable stiffness optimization design of macro and micro, the macro structure topology design variable is introduced, and the calculation formula of the unit elastic constitutive matrix is:

[0083] ;

[0084] wherein, is the unit elastic constitutive matrix of the th unit, is the macro structure topology optimization design variable, the range of which is , is the macro topology optimization design variable penalty coefficient.

[0085] In summary, the element elastic constitutive matrix under the macro- and micro-level concurrent multi-scale variable stiffness optimization design framework... , can be represented as:

[0086] ;

[0087] S6: Calculate the element stiffness based on the element elastic constitutive matrix, and assemble the overall stiffness based on the element stiffness.

[0088] The element stiffness is calculated as follows:

[0089] ;

[0090] in, For the first The element stiffness matrix of each element. This is the strain-displacement matrix, with superscripts. To represent the transpose of a matrix, This is a unit domain.

[0091] The static equilibrium equations of the structure are expressed as follows:

[0092] ;

[0093] in, This is the structural load vector.

[0094] S7: Based on the overall stiffness, perform static equilibrium analysis of the structure and obtain the objective function;

[0095] With structural flexibility Minimize to the objective function, as shown in the formula:

[0096] ;

[0097] in, Let be the overall displacement vector of the structure. This is the overall stiffness matrix of the structure.

[0098] S8: Calculate the volume constraints of the design region;

[0099] The formula for calculating the volume constraint is:

[0100] ;

[0101] in, The actual volume of the structure. For the first The unit volume of each cell. This represents the total number of units.

[0102] S9: Perform volume constraint and objective function sensitivity analysis;

[0103] The volume constraint and the sensitivity analysis of the objective function in S9 are performed, including:

[0104] Macro-structure topology optimization design variable The sensitivity of the objective function is:

[0105] ;

[0106] The total stiffness matrix of the structure The partial derivative of the design variable can be calculated separately for the first element stiffness matrix , because each element corresponds to an independent design variable:

[0107] ;

[0108] ;

[0109] Where, is the area to be designed;

[0110] Micro-fiber laying angle design variable , The sensitivity of the objective function , only need to calculate , :

[0111] The calculation is:

[0112] ;

[0113] ;

[0114] ;

[0115] The calculation is:

[0116] ;

[0117] ;

[0118] ;

[0119] Macro-structure topology optimization design variable , micro-fiber laying angle design variable , The sensitivity of the volume constraint , ,​ For:

[0120] , , ;

[0121] S10: enter the optimization iteration loop, repeat steps S4-S9 until the convergence criterion is met, and realize the three-dimensional fiber-reinforced composite material multi-scale variable stiffness optimization design.

[0122] The convergence criterion is:

[0123] ;

[0124] Wherein, is the convergence criterion, is the convergence tolerance, is a positive integer, indicating the number of consecutive iteration steps, is the current iteration step, is the th consecutive iteration step.

[0125] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be further described in detail below in combination with the drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application and do not limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other. It should be noted that only numerical examples of three-dimensional cantilever beams are given in the present embodiment. It does not mean that the present application cannot be extended to other arbitrary structures.

[0126] As Figure 2 shown, given a three-dimensional cantilever beam, and apply a response load, obtain the three-dimensional fiber-reinforced composite material multi-scale variable stiffness optimization design result as shown in Figure 3 . In summary, the present numerical example fully demonstrates that the present application is effective.

Claims

1. A multi-scale variable stiffness optimization design method for three-dimensional fiber-reinforced composite materials, characterized in that, Includes the following steps: S1: Perform finite element mesh generation on the structure to be designed to obtain... Design unit; S2: Select the number of alternative rotation angle 1 and alternative rotation angle 2; S3: Initialize macroscopic structural topology optimization and microscopic fiber laying angle design variables; S4: Calculate the micro-elastic constitutive matrix based on the two-dimensional normal distribution fiber optimization interpolation scheme and the micro-fiber laying angle design variable; S5: Calculate the unit elastic constitutive matrix based on the micro-elastic constitutive matrix and macro-structural topology optimization design variables; S6: Calculate the element stiffness based on the element elastic constitutive matrix, and assemble the overall stiffness based on the element stiffness. S7: Based on the overall stiffness, perform static equilibrium analysis of the structure and obtain the objective function; S8: Calculate the volume constraints of the design region; The formula for calculating the volume constraint in S8 is as follows: ； in, The actual volume of the structure. For the first The unit volume of each cell. The total number of units, The design variables for macroscopic structural topology optimization are within the following range. ; S9: Perform volume constraint and objective function sensitivity analysis; The volume constraint and objective function sensitivity analysis are performed in S9, including: Macroscopic structural topology optimization design variables For the objective function The sensitivity is: ； in, Let be the overall displacement vector of the structure. This is the overall stiffness matrix of the structure, with superscript... Represents the transpose of a matrix; Overall structural stiffness matrix The partial derivatives with respect to the design variables are obtained by applying the element stiffness matrix. Perform the calculation: ； ； in, For the area that needs to be designed, The strain-displacement matrix, For the first The element elastic constitutive matrix of each element. Design penalty coefficients for macroscopic topology optimization variables. and These represent the number of alternative rotation angle 1 and alternative rotation angle 2, respectively. For the normalized first The weighting coefficients of the elastic constitutive matrix of the candidate fiber layup angles for each unit. For the first species, first Elastic constitutive matrix of alternative fiber layup angles; Microfiber layup angle design variables , For the objective function The sensitivity only needs to be calculated. , : The calculation is as follows: ； ； ； in, For the first The weighting coefficients of the elastic constitutive matrix of the candidate fiber layup angles for each unit. For angle Optimize the penalty coefficient. For angle Optimize penalty coefficient; The calculation is as follows: ； ； ； Macroscopic structural topology optimization design variables Microfiber laying angle design variables , Volume constraints Sensitivity , , for: ， ， ； S10: Enter the optimization iteration loop, repeat steps S4-S9 until the convergence criterion is met, and realize the multi-scale variable stiffness optimization design of three-dimensional fiber reinforced composite materials.

2. The multi-scale variable stiffness optimization design method for three-dimensional fiber-reinforced composite materials according to claim 1, characterized in that, The formula for calculating the microelastic constitutive matrix in S4 is as follows: ； ； ； in, For the first The microelastic constitutive matrix of each unit, For the first The angle of each unit Design variables, the range of which is , For the first The angle of each unit Design variables, the range of which is .

3. The multi-scale variable stiffness optimization design method for three-dimensional fiber-reinforced composite materials according to claim 2, characterized in that, The formula for calculating the elastic constitutive matrix of the element in S5 is as follows: 。 4. The multi-scale variable stiffness optimization design method for three-dimensional fiber-reinforced composite materials according to claim 3, characterized in that, The element stiffness calculation in S6 is as follows: ； in, This is a unit domain.

5. The multi-scale variable stiffness optimization design method for three-dimensional fiber-reinforced composite materials according to claim 4, characterized in that, S7 emphasizes structural flexibility Minimize to the objective function, as shown in the formula: 。 6. The multi-scale variable stiffness optimization design method for three-dimensional fiber-reinforced composite materials according to claim 5, characterized in that, The convergence criterion in S10 is: ； in, For the convergence criterion, To reduce convergence tolerance, A positive integer, representing a continuous sequence. One iteration step, For the current iteration step, For the first One consecutive iteration step.

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