A discrete-continuous optimization variable interpolation calculation method

Abstract

The present application belongs to the field of joint optimization of anisotropic material direction and topology, and particularly relates to a discrete continuous optimization variable interpolation calculation method, which converts the optimization of material direction angle into four intervals according to the period of the material direction conversion matrix, and calculates the weight function of the material stiffness of the four intervals; defines the initial stiffness of the anisotropic material, and then determines the material stiffness of the four intervals in combination with the weight function; calculates the macroscopic structure stiffness according to the material stiffness of the four intervals; obtains the relationship between the real material direction after optimization and the discrete variables and the continuous variables based on the continuous variables; and provides a new discrete continuous interpolation method with the help of the concept of weight function in the finite element method when performing joint optimization of anisotropic material direction and topology, which can reduce design variables and improve optimization efficiency on the one hand; and prevent the material direction from falling into the local optimal solution on the other hand, and is used for the collaborative optimization design of material direction and topology of composite material structures.

Description

Technical Field

[0001] The present application belongs to the field of joint optimization of anisotropic material orientation and topology, and particularly relates to a discrete continuous optimization variable interpolation calculation method. Background Art

[0002] When performing joint optimization of anisotropic material direction and topology, the material direction conversion matrix calculation, that is, the continuous material direction angle optimization problem, is involved. This problem can easily fall into the local optimal solution during the material direction angle interval optimization process. Studies have found that the above problem can be solved by discrete interval technology, but related research has many discrete related design variables, which has a great impact on the efficiency of topology optimization. This application uses the concept of weight function in the finite element method to provide a new discrete continuous interpolation method for the collaborative optimization design problem of material direction and topology. Summary of the invention

[0003] The purpose of this application is to provide a discrete continuous optimization variable interpolation calculation method to solve the problem that there are many existing discrete related design variables and that they have a great impact on the efficiency of topology optimization.

[0004] The technical solution of the present application is: a discrete continuous optimization variable interpolation calculation method, comprising:

[0005] Determine the period of the material direction transformation matrix and convert the material direction angle The optimization is converted into four intervals, and the material direction conversion matrix is ​​established;

[0006] The shape function is introduced as the weight function to calculate the weight function of the material stiffness in four intervals;

[0007] Define the initial stiffness of the anisotropic material, and then determine the material stiffness of the four intervals in combination with the material direction transformation matrix;

[0008] The macroscopic structural stiffness is calculated based on the material stiffness and weight functions in four intervals;

[0009] Based on continuous variables, the relationship between the actual material direction after optimization and the discrete and continuous variables is obtained;

[0010] The macroscopic structural stiffness of the four intervals is obtained, and the mathematical model and optimization formula of topology optimization are established by combining the real material direction with the relationship between discrete variables and continuous variables. Then, the objective function based on strain energy is set, with the minimum strain energy of the objective function as the goal, and the discrete variables, continuous variables and material density in the objective function are iteratively updated to obtain the optimal material direction and topological shape, and complete the interpolation calculation.

[0011] Preferably, the period of the material direction conversion matrix is , the material direction angle The optimization is converted into four intervals, each interval size is , the material direction transformation matrix is ​​obtained as:

[0012] .

[0013] Preferably, the weight function satisfies , weight functions of material stiffness in four intervals , , and They are:

[0014] ;

[0015] ;

[0016] ;

[0017] ;

[0018] In the formula, and All are discrete variables.

[0019] Preferably, the initial stiffness of the anisotropic material is Defined as:

[0020] ;

[0021] In the formula, , , , , are the elastic constants in different directions of anisotropic materials;

[0022] Material stiffness in four intervals , , and They are:

[0023] ;

[0024] Where T is the material direction transformation matrix, T T is the transpose of the material orientation transformation matrix.

[0025] Preferably, the macroscopic structural stiffness for:

[0026] ;

[0027] In the formula, is a continuous variable, The material information of each interval is superimposed in the form of a continuous weight function. for: , The range length is .

[0028] Preferably, the actual material direction after optimization is , and The relationship between them is:

[0029] ;

[0030] In the formula, is a continuous variable, is the actual material direction after optimization.

[0031] Preferably, the material interpolation calculation method is used to establish the topology optimization mathematical model and optimization formula, and the discrete variables in the objective function are iteratively updated using the gradient descent method. , continuous variables and material density .

[0032] The discrete continuous optimization variable interpolation calculation method of the present application provides a new discrete continuous interpolation method when performing joint optimization of anisotropic material direction and topology with the help of the concept of weight function in the finite element method. On the one hand, it can reduce design variables and improve optimization efficiency; on the other hand, it can prevent the material direction from falling into the local optimal solution, and is used for the collaborative optimization design of material direction and topology of composite materials. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] In order to more clearly illustrate the technical solution provided by the present application, the following is a brief introduction to the accompanying drawings. Obviously, the accompanying drawings described below are only some embodiments of the present application.

[0034] Figure 1 Flowchart for the collaborative optimization of anisotropic material orientation and topology;

[0035] Figure 2 This is the overall flow chart of this application;

[0036] Figure 3 A schematic diagram of the discrete optimization of the direction angle of this application material in the basic coordinate system;

[0037] Figure 4 This is a schematic diagram of the application effect of the L-beam verification example of this application;

[0038] Figure 5 This is a schematic diagram of the application effect of the simply supported beam verification example in this application. DETAILED DESCRIPTION

[0039] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0040] A discrete continuous optimization variable interpolation calculation method, such as Figure 1 As shown, it is a flow chart of the coordinated optimization of anisotropic material direction and topology. The present application is mainly used for the interpolation calculation therein, and discrete continuous operations of the material direction are performed in this link, including the conversion of the material direction and the combination with the optimization design variables.

[0041] Anisotropic materials have the property that all or part of their physical, chemical and other properties in different directions show certain differences depending on the direction, that is, the performance values ​​measured in different directions are different.

[0042] like Figure 2 As shown, the following steps are included:

[0043] Step S100, determining the period of the material direction conversion matrix, converting the material direction angle The optimization is converted into four intervals, and the material direction conversion matrix is ​​established;

[0044] The material direction transformation matrix is ​​defined as follows. It can be directly seen that the period of the material direction transformation matrix is , if the gradient-based algorithm is used to optimize the material direction angle in topology optimization, it is easy to obtain a local optimal solution. Figure 3 As shown in ①-④, the material direction angle The optimization is converted into four intervals, each interval is only , which can greatly reduce the above risks; Figure 3 The dashed line direction is the orthotropic material direction.

[0045] The material direction transformation matrix T is:

[0046] .

[0047] Step S200: introducing the shape function as a weight function, and calculating the weight functions of the material stiffness of four intervals.

[0048] Preferably, the weight function satisfies , weight functions of material stiffness in four intervals , , and They are:

[0049] ;

[0050] ;

[0051] ;

[0052] ;

[0053] In the formula, and All are discrete variables.

[0054] Step S300, defining the initial stiffness of the anisotropic material, and then determining the material stiffness of four intervals in combination with the material direction conversion matrix.

[0055] Anisotropic material initial stiffness Defined as:

[0056] ;

[0057] In the formula, , , , , They are the elastic constants in different directions of anisotropic materials, which are calculated from the engineering constants Young's modulus and Poisson's ratio.

[0058] Material stiffness in four intervals , , and They are:

[0059] ;

[0060] Where T is the material direction transformation matrix, T T is the transpose of the material orientation transformation matrix.

[0061] Step S400, calculating the macroscopic structural stiffness according to the material stiffness and weight function of the four intervals.

[0062] Macrostructural stiffness for:

[0063] ;

[0064] In the formula, is a continuous variable, The material information of each interval is superimposed in the form of a continuous weight function. for: , The range length is .

[0065] Step S500, obtaining the actual material direction and , and The relationship between them is:

[0066] ;

[0067] In the formula, is the actual material direction after optimization.

[0068] Step S600, obtain the macroscopic structural stiffness of the four intervals, and combine the actual material direction with , and The relationship between the topology optimization mathematical model and optimization formula are established by using the material interpolation calculation method; then the objective function based on strain energy is set , with the objective function The goal is to minimize the strain energy, and the discrete variables in the objective function are iteratively updated using the gradient descent method. , continuous variables and material density , get the optimal material direction and topological shape, and complete the interpolation calculation. Figure 4 and Figure 5 As shown, Figure 4 Verify the application effect of the example for L-beam. Figure 5 Verify the application effect of the example for a simply supported beam.

[0069] In summary, this application provides a new discrete continuous interpolation method with the help of the concept of weight function in the finite element method when performing joint optimization of anisotropic material direction and topology. On the one hand, it can reduce design variables and improve optimization efficiency; on the other hand, it can prevent the material direction from falling into the local optimal solution, which is used for the collaborative optimization design of material direction and topology of composite materials.

[0070] Finally, it should be noted that: the drawings of the embodiments disclosed in the present invention only involve structures related to the embodiments disclosed in the present invention, and other structures can refer to the general design. In the absence of conflict, the same embodiment and different embodiments of the present invention can be combined with each other;

[0071] Finally: The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A discrete continuous optimization variable interpolation calculation method, characterized in that: include: Determine the period of the material direction conversion matrix, convert the optimization of the material direction angle θ into four intervals according to the period of the material direction conversion matrix, and establish the material direction conversion matrix; The shape function is introduced as the weight function to calculate the weight function of the material stiffness in four intervals; Define the initial stiffness of the anisotropic material, and then determine the material stiffness of the four intervals in combination with the material direction transformation matrix; The macroscopic structural stiffness is calculated based on the material stiffness and weight functions in four intervals; Based on continuous variables, the relationship between the actual material direction after optimization and the discrete and continuous variables is obtained; The macroscopic structural stiffness of the four intervals is obtained, and the relationship between the real material direction and discrete variables and continuous variables is combined to establish a mathematical model and optimization formula for topology optimization. Then, a strain energy-based objective function is set, with the minimum strain energy of the objective function as the goal, and the discrete variables, continuous variables and material density in the objective function are iteratively updated to obtain the optimal material direction and topological shape, completing the interpolation calculation. The weight function satisfies The weight functions W1, W2, W3 and W4 of the material stiffness in the four intervals are: In the formula, r and s are both discrete variables; The initial stiffness D of the anisotropic material i Defined as: Where D 11 , D 12 , D 21 , D 22 , D 66 are the elastic constants in different directions of anisotropic materials; The material stiffnesses D1, D2, D3 and D4 in the four intervals are: Where T is the material direction transformation matrix, T T is the transpose of the material orientation transformation matrix.

2. The discrete continuous optimization variable interpolation calculation method according to claim 1, characterized in that: The period of the material direction conversion matrix is ​​π, and the optimization of the material direction angle θ is converted into four intervals, each of which is The material direction transformation matrix is ​​obtained as:

3. The discrete continuous optimization variable interpolation calculation method according to claim 1, characterized in that: The macrostructural stiffness D macro for: D macro =T T (θ c )D con T(θ c ); In the formula, θ c is a continuous variable, D con The material information of each interval is superimposed in the form of a continuous weight function, D con for: θ c The range length is 4. The discrete continuous optimization variable interpolation calculation method according to claim 1, characterized in that: The actual material direction after optimization is related to r, s and θ c The relationship between them is: In the formula, θ c is a continuous variable, θ real is the actual material direction after optimization.

5. The discrete continuous optimization variable interpolation calculation method according to claim 1, characterized in that: The material interpolation calculation method is used to establish the topology optimization mathematical model and optimization formula, and the gradient descent method is used to iteratively update the discrete variables r, s and continuous variables θ in the objective function. c and material density P.

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