Variable-stiffness multi-point shape-preserving multi-scale optimization design method for fiber reinforced composite material

Abstract

The invention discloses a multi-point shape-preserving multi-scale optimization design method for variable stiffness of a fiber reinforced composite material, and relates to the field of variable stiffness design of the fiber reinforced composite material. Based on a normal distribution fiber optimization interpolation format, a multi-point shape-preserving design and an artificial weak unit concept are introduced; a fiber reinforced composite material multi-point shape-preserving multi-scale variable stiffness optimization design research is developed, and the analysis sensitivity of a design variable to multi-point shape-preserving constraint is deduced through an adjoint vector method. The method solves the following difficulties in the prior art: (1) the calculation time consumption is huge due to the increase of the design variables along with the grid density and the number of alternative fiber laying angles; (2) adopting the minimum displacement of all nodes in a local area as a design target to cause an unfavorable fixed area with a pure rigid shape and a fixed position; (3) the perforated structure does not have actual physical elements for calculating the local deformation energy of the structure; and (4) after the artificial weak unit concept is introduced, the analysis sensitivity derivation is difficult.

Description

Technical Field

[0001] The invention relates to the field of variable stiffness design of fiber reinforced composite materials, and in particular to a variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials. Background Art

[0002] With the growing demand for lightweight components in the aerospace, automotive and transportation industries, fiber-reinforced composites have become the preferred material for the engineering community to solve the challenges of lightweighting and performance optimization due to their excellent properties such as designability, high specific strength, high specific modulus and corrosion resistance. In aerospace equipment, there are many open structures, such as holes for the installation and connection of fasteners, ventilation systems and other accessories. These hole structures may cause problems such as stress concentration, and the size of their deformation directly determines the performance and safety of the overall structure. Therefore, it is crucial to study the deformation control of the open-porous structure of fiber-reinforced composites. Based on the concept of concurrent optimization of structure / materials, multi-scale variable stiffness conformal optimization design of composite materials for open-porous structures is an effective way to achieve structural lightweighting while controlling the deformation of open-porous structures. There are mainly the following difficulties in carrying out multi-scale variable stiffness conformal optimization design of composite materials: 1) The design variables increase linearly with the increase of grid density and the number of alternative fiber laying angles, which will lead to a huge calculation time; 2) If the minimum displacement of all nodes in the local area is directly adopted as the design target, it will lead to an unfavorable pure rigid shape and a fixed area with fixed position; 3) The open-pore structure does not have actual physical elements for the calculation of the local deformation energy of the structure; 4) Due to the introduction of the concept of artificial weak elements, it is difficult to derive the analytical sensitivity. Summary of the invention

[0003] In view of the above-mentioned deficiencies in the prior art, the present invention provides a variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials.

[0004] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is: a variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials, comprising the following steps: S1: Divide the fiber reinforced composite structure into regions, including dividing the area to be designed into the design domain and the structural opening area into the conformal domain; S2: Divide the design domain into finite element units and divide it into design units; S3: Perform artificial weak unit division on the conformal domain and divide it into An artificial weak unit; S4: Initialize macro-design variables and micro-design variables; S5: Based on the initialization results, the volume constraint calculation of the design domain is performed; S6: Perform multi-point conformal design constraint calculations on the conformal domain; S7: Perform finite element calculation on the overall design structure and obtain the objective function; S8: Perform volume constraint sensitivity analysis, multi-point conformal design constraint sensitivity analysis, and objective function sensitivity analysis; S9: Based on the sensitivity analysis results, determine whether the convergence standard is reached. If so, complete the variable stiffness multi-point conformal multi-scale optimization design of the fiber reinforced composite material, otherwise return to step S5.

[0005] Furthermore, in S5, the volume constraint calculation of the design domain is performed, and the formula is:

[0006]

[0007]

[0008] in, Design domain The volume of is the initial given volume fraction, Design domain The volume and For the The volume of a design unit, is the macro design variable, is the unit area, is the thickness of the laminate.

[0009] Furthermore, in S6, a multi-point conformal design constraint calculation is performed on the conformal domain, and the formula is:

[0010]

[0011]

[0012] in, Conformal Domain The strain energy, To set the parameters, Conformal Domain The warping deformation, Conformal Domain The stiffness matrix of Represents the transpose of a matrix.

[0013] Furthermore, in S7, finite element calculation is performed on the overall design structure, and the formula is:

[0014] in, For laminate Tier The element stiffness matrix of the element, is the strain-displacement matrix, Design domain Elastic constitutive matrix of meso-macroscopic candidate materials, Design variables for micro level; The objective function is to minimize the structural flexibility:

[0015]

[0016] in, For structural flexibility, is the overall displacement, is the overall stiffness, is the overall external load vector.

[0017] Furthermore, the volume constraint sensitivity analysis in S8 is:

[0018] in, It means partial derivative.

[0019] Furthermore, the multi-point conformal design constraint sensitivity analysis in S8 is: Multi-point conformal design constraints on macro design variables The sensitivity is:

[0020] Multi-point conformal design constraints on micro-design variables The sensitivity is:

[0021] in, is the artificial displacement vector.

[0022] Furthermore, the sensitivity analysis of the objective function in S8 is: Objective function for macro design variables The sensitivity is:

[0023] in By individually Tier The element stiffness matrix of the element Perform the calculation:

[0024]

[0025] in, is the macro penalty parameter, is the elastic constitutive matrix of the micro-element; Objective function for micro-design variables The sensitivity is: .

[0026] Furthermore, the convergence criterion in S9 is:

[0027] in, is the convergence criterion, For the Step objective function, For the Step objective function, is the maximum objective function in the iteration, is the minimum objective function in the iteration, is the convergence tolerance.

[0028] The beneficial effects of the present invention are: (1) The present invention clearly presents the derivation of volume constraints, multi-point conformal constraints, and analytical sensitivity of the objective function; (2) The present invention solves the difficulty of having no actual physical elements for calculating the local deformation energy of the open hole structure through the concept of artificial weak elements; (3) The present invention is applicable to any fiber-reinforced composite material with any open-pore structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 Flowchart of the variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials.

[0030] Figure 2 Schematic diagram of the multi-point conformal design and artificial weak unit concept of the present invention.

[0031] Figure 3 Schematic diagram of the calculation structure of an L-beam with a square hole.

[0032] Figure 4 Schematic diagram of variable stiffness multi-scale optimization design results for ordinary fiber-reinforced composite materials.

[0033] Figure 5 It is a schematic diagram of the variable stiffness multi-point conformal multi-scale optimization design results of the fiber reinforced composite material of the present invention. DETAILED DESCRIPTION

[0034] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0035] Based on the normally distributed fiber optimization interpolation format, the present invention introduces the concepts of multi-point conformal design and artificial weak units, carries out research on multi-point conformal and multi-scale variable stiffness optimization design of fiber-reinforced composite materials, derives the analytical sensitivity of design variables to multi-point conformal constraints through the adjoint vector method, and proposes a variable stiffness multi-point conformal and multi-scale optimization design method for fiber-reinforced composite materials.

[0036] like Figure 1 As shown, a variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials includes the following steps: S1: Divide the fiber reinforced composite structure into regions, including dividing the area to be designed into the design domain and the structural opening area into the conformal domain; S2: Divide the design domain into finite element units and divide it into design units; S3: Perform artificial weak unit division on the conformal domain and divide it into An artificial weak unit; S4: Initialize macro-design variables and micro-design variables; S5: Based on the initialization results, the volume constraint calculation of the design domain is performed; S6: Perform multi-point conformal design constraint calculations on the conformal domain; S7: Perform finite element calculation on the overall design structure and obtain the objective function; S8: Perform volume constraint sensitivity analysis, multi-point conformal design constraint sensitivity analysis, and objective function sensitivity analysis; S9: Based on the sensitivity analysis results, determine whether the convergence standard is reached. If so, complete the variable stiffness multi-point conformal multi-scale optimization design of the fiber reinforced composite material, otherwise return to step S5.

[0037] When performing conformal design on the hole area, the stiffness matrix of the conformal area is not calculated. Therefore, the present invention introduces the concept of multi-point conformal design and artificial weak unit, such as Figure 2 As shown in Figure 2, the main idea is to consider multiple key points on the hole edge that need to control deformation and establish An artificial weak unit.

[0038] In S5, the volume constraint calculation of the design domain is performed, and the formula is:

[0039]

[0040]

[0041] in, Design domain The volume of is the initial given volume fraction, Design domain The volume and For the The volume of a design unit, is the macro design variable, is the unit area, is the thickness of the laminate.

[0042] Structural conformal domain Under load The total deformation under the action is expressed as the node displacement vector in the region , which includes the rigid body displacement and warpage , specifically expressed as:

[0043] Since conformal design is to achieve the effect of coordinated displacement without warping, the rigid body displacement of the structure is not considered in the present invention. The degree of warping can be quantitatively described using regional deformation.

[0044] In order to prevent numerical singularity, the artificial weak element is given very small material properties so that the corresponding regional strain energy can be calculated. .

[0045] Considering that the actual engineering problem cannot achieve the ideal conformal design effect ( ), so given , which is used to limit the degree of warping deformation in the opening area.

[0046] In S6, multi-point conformal design constraint calculation is performed on the conformal domain, and the formula is:

[0047]

[0048]

[0049] in, Conformal Domain The strain energy, To set the parameters, Conformal Domain The warping deformation, Conformal Domain The stiffness matrix of Represents the transpose of a matrix.

[0050] The structure is subjected to overall finite element calculations, and the static response of the multi-scale composite laminate is obtained by using the first-order shear deformation theory and assuming that the structure is linear elastic. Tier The element stiffness array of the elements The design domain can be Elastic constitutive matrix of meso-macroscopic candidate materials Points earned.

[0051] The linear static equilibrium equation of the structure is expressed as:

[0052] In S7, the finite element calculation of the overall design structure is performed, and the formula is:

[0053] in, For laminate Tier The element stiffness matrix of the element, is the strain-displacement matrix, Design domain Elastic constitutive matrix of meso-macroscopic candidate materials, Design variables for micro level; The objective function is to minimize the structural flexibility:

[0054]

[0055] in, For structural flexibility, is the overall displacement, is the overall stiffness, is the overall external load vector.

[0056] The present invention is based on a multi-point conformal multi-scale variable stiffness optimization framework for fiber reinforced composite materials, taking the overall material usage and the deformation energy of the hole area as constraints, and taking the minimum structural compliance as the objective function. The mathematical formula of the optimization problem is expressed as follows:

[0057]

[0058]

[0059] in, is the set of design variables, Types of fiber laying angles for selection.

[0060] The volume constraint sensitivity analysis in S8 is:

[0061] in, It means partial derivative.

[0062] The multi-point conformal design constraint sensitivity analysis in S8 is: Multi-point conformal design constraints on macro design variables The sensitivity is calculated as:

[0063] in is a symmetric matrix, , the above formula can be combined to express as:

[0064] Nodal displacement of artificial weak element The coefficient matrix Overall displacement of the structure Extracted from:

[0065] Establishing the node displacement of artificial weak element right Partial derivative and overall displacement right The relationship between the partial derivatives of is:

[0066] Substitute into the structural static analysis and integrate into:

[0067] Since the structural stiffness matrix is ​​directly It is difficult to find the inverse matrix, so the adjoint method is used to continue the derivation, let:

[0068] Will It is applied as an artificial load vector to the overall structure and a finite element analysis is performed to obtain an artificial displacement vector :

[0069] Integrating the above derivation process, it can be expressed as:

[0070] Multi-point conformal design constraints on macro design variables The sensitivity is:

[0071] Multi-point conformal design constraints on micro-design variables The sensitivity is:

[0072] in, is the artificial displacement vector, , Both are 0.

[0073] The sensitivity analysis of the objective function in S8 is: Objective function for macro design variables The sensitivity is:

[0074] in By individually Tier The element stiffness matrix of the element Perform the calculation:

[0075]

[0076] in, is the macro penalty parameter, is the elastic constitutive matrix of the micro-element; Objective function for micro-design variables The sensitivity is:

[0077] The convergence criterion in S9 is:

[0078] in, is the convergence criterion, For the Step objective function, For the Step objective function, is the maximum objective function in the iteration, is the minimum objective function in the iteration, is the convergence tolerance.

[0079] In one embodiment of the present invention, Figure 3 As shown, given an L-beam structure with a square hole, and applying a response load, such as Figure 4 and Figure 5As shown, by comparing the results obtained under the ordinary scheme and the scheme provided by the present invention, it is found that the objective functions without applying and applying multi-point conformal constraints are 157.49 and 169.21 respectively. Applying multi-point conformal constraints sacrifices 7.44% of the objective function. After applying multi-point conformal constraints, the absolute displacement of the four key points becomes larger, but the relative displacement becomes smaller, especially key point 12 (26.19% in the X direction), key point 23 (43.75% in the X direction), and key point 34 (27.82% in the X direction). From the perspective of macroscopic configuration, after applying multi-point conformal constraints, the lower left corner structure is closer to hinge. After applying multi-point conformal constraints, there is also a situation where the relative displacement between key points becomes larger, for example, key point 14 (0.886% in the X direction and 139.25% in the Y direction), but the absolute displacement between these two points is small. It is believed that the displacement of this position is sacrificed during the optimization process to coordinate the relative displacement between key points 12, 23, and 34. In summary, this numerical example fully illustrates that the proposal of the present invention is effective.

[0080] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the invention.

Claims

1. A variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials, characterized in that: The following steps are involved: S1: Divide the fiber reinforced composite structure into regions, including dividing the area to be designed into the design domain and the structural opening area into the conformal domain; S2: Divide the design domain into finite element units and divide it into design units; S3: Perform artificial weak unit division on the conformal domain and divide it into An artificial weak unit; S4: Initialize macro-design variables and micro-design variables; S5: Based on the initialization results, the volume constraint calculation of the design domain is performed; S6: Perform multi-point conformal design constraint calculations on the conformal domain; S7: Perform finite element calculation on the overall design structure and obtain the objective function; S8: Perform volume constraint sensitivity analysis, multi-point conformal design constraint sensitivity analysis, and objective function sensitivity analysis; S9: Based on the sensitivity analysis results, determine whether the convergence standard is reached. If so, complete the variable stiffness multi-point conformal multi-scale optimization design of the fiber reinforced composite material, otherwise return to step S5.

2. The variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials according to claim 1, characterized in that: In S5, the volume constraint calculation of the design domain is performed, and the formula is: in, Design domain The volume of is the initial given volume fraction, Design domain The volume and For the The volume of a design unit, is the macro design variable, is the unit area, is the thickness of the laminate.

3. The variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials according to claim 2, characterized in that: In S6, multi-point conformal design constraint calculation is performed on the conformal domain, and the formula is: in, Conformal Domain The strain energy, To set the parameters, Conformal Domain The warping deformation, Conformal Domain The stiffness matrix of Represents the transpose of a matrix.

4. The variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials according to claim 3, characterized in that: In S7, the finite element calculation of the overall design structure is performed, and the formula is: in, For laminate Tier The element stiffness matrix of the element, is the strain-displacement matrix, Design domain Elastic constitutive matrix of meso-macroscopic candidate materials, Design variables for micro level; The objective function is to minimize the structural flexibility: in, For structural flexibility, is the overall displacement, is the overall stiffness, is the overall external load vector.

5. The variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials according to claim 4, characterized in that: The volume constraint sensitivity analysis in S8 is: in, It means partial derivative.

6. The variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials according to claim 5, characterized in that: The multi-point conformal design constraint sensitivity analysis in S8 is: Multi-point conformal design constraints on macro design variables The sensitivity is: Multi-point conformal design constraints on micro-design variables The sensitivity is: in, is the artificial displacement vector.

7. The variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials according to claim 6, characterized in that: The sensitivity analysis of the objective function in S8 is: Objective function for macro design variables The sensitivity is: in By individually Tier The element stiffness matrix of the element Perform the calculation: in, is the macro penalty parameter, is the elastic constitutive matrix of the micro-element; Objective function for micro-design variables The sensitivity is: 。 8. The variable stiffness multi-point conformal multi-scale optimization design method for fiber reinforced composite materials according to claim 7, characterized in that: The convergence criterion in S9 is: in, is the convergence criterion, For the Step objective function, For the Step objective function, is the maximum objective function in the iteration, is the minimum objective function in the iteration, is the convergence tolerance.

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