Multi-point shape-preserving multi-scale optimization design method for fiber reinforced composites with variable stiffness

Abstract

The present invention discloses a multi-point conformal multi-scale optimization design method for variable stiffness of fiber reinforced composite materials, which relates to the field of variable stiffness design of fiber reinforced composite materials. The present invention is based on the normal distribution fiber optimization interpolation format, introduces the concept of multi-point conformal design and artificial weak unit, carries out research on multi-point conformal multi-scale variable stiffness optimization design of fiber reinforced composite materials, and derives the analytical sensitivity of design variables to multi-point conformal constraints by adjoint vector method. The present invention solves the following difficulties of the prior art: (1) the design variables become time-consuming to calculate as the grid density and the number of alternative fiber laying angles increase; (2) the use of the minimum displacement of all nodes in the local area as the design target will lead to an unfavorable pure rigid shape and a fixed area of ​​fixed position; (3) the open-cell structure has no 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 unit, the analytical sensitivity derivation is difficult.

Description

Technical Field

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

[0002] With the continuous growth of the demand for lightweight components in the aerospace, automotive, and transportation industries, fiber - reinforced composites have become the preferred materials for the engineering community to address 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 often open - hole structures, such as installation and connection holes for fasteners, ventilation systems, and other accessories. These hole structures may cause problems such as stress concentration, and their deformation magnitudes directly determine the performance and safety of the overall structure. Therefore, the research on deformation control of open - hole structures in fiber - reinforced composites is of crucial importance. Based on the concept of concurrent structure / material optimization, for open - hole structures, carrying out multi - scale variable stiffness conformal optimization design of composites is an effective way to control the deformation of open - hole structures while achieving structural lightweighting. There are mainly the following difficulties in carrying out multi - scale variable stiffness conformal optimization design of composites: 1) The design variables increase linearly with the increase in mesh density and the number of alternative fiber laying angles, which will lead to huge computational time consumption; 2) If directly using the minimum displacement of all nodes in a local area as the design goal, it will lead to an unfavorable pure rigid shape and a fixed area with immovable positions; 3) There are no actual physical elements in open - hole structures for calculating the local deformation energy of the structure; 4) Due to the introduction of the artificial weak element concept, it is difficult to deduce the analytical sensitivity. Summary of the Invention

[0003] Aiming at the above deficiencies in the prior art, the present invention provides a multi - point conformal multi - scale optimization design method for variable stiffness of fiber - reinforced composites.

[0004] To achieve the above - mentioned invention objective, the technical solution adopted by the present invention is as follows: A multi - point conformal multi - scale optimization design method for variable stiffness of fiber - reinforced composites, comprising the following steps:

[0005] S1: Divide the fiber - reinforced composite material structure into regions, including dividing the region to be designed into a design domain and the structure open - hole region into a conformal domain;

[0006] S2: Conduct finite - element cell division on the design domain and divide it into design cells;

[0007] S3: Conduct artificial weak - element division on the conformal domain and divide it into artificial weak elements;

[0008] S4: Initialize the macro design variables and micro design variables;

[0009] S5: Based on the initialization result, perform volume constraint calculation on the design domain;

[0010] S6: Perform multi-point conformal design constraint calculation on the conformal domain;

[0011] S7: Perform finite element calculation on the overall design structure and obtain the objective function;

[0012] S8: Conduct volume constraint sensitivity analysis, multi-point conformal design constraint sensitivity analysis, and objective function sensitivity analysis;

[0013] S9: Based on the sensitivity analysis result, determine whether the convergence criterion is met. If so, complete the variable stiffness multi-point conformal multi-scale optimization design of the fiber-reinforced composite material; otherwise, return to step S5.

[0014] Further, in S5, the volume constraint calculation on the design domain is performed according to the formula:

[0015]

[0016]

[0017]

[0018] Where, is the volume of the design domain ; is the initially given volume fraction, is the sum of the volumes of the design domain ; is the volume of the th design unit, are the macro design variables, is the unit area, is the laminate thickness.

[0019] Further, in S6, the multi-point conformal design constraint calculation on the conformal domain is performed according to the formula:

[0020]

[0021]

[0022]

[0023] Where, is the strain energy of the conformal domain ; is the set parameter, is the conformal domain Warpage deformation For the conformal domain The stiffness matrix, with the superscript Indicating the transpose of the matrix

[0024] Furthermore, in S7, a finite element calculation is performed on the overall design structure, and the formula is:

[0025]

[0026] Where Is the element stiffness matrix of the th layer and the th element of the laminated plate, Is the strain-displacement matrix, Is the design domain The elastic constitutive matrix of the macroscopic alternative materials in, Is the microscopic design variable;

[0027] Taking the minimum structural compliance as the objective function:

[0028]

[0029]

[0030] Where Is the structural compliance, Is the total displacement, Is the total stiffness, Is the total external load vector

[0031] Furthermore, the volume constraint sensitivity analysis in S8 is:

[0032]

[0033] Where Indicates taking the partial derivative

[0034] Furthermore, the multi-point conformal design constraint sensitivity analysis in S8 is:

[0035] The sensitivity of the multi-point conformal design constraint to the macroscopic design variable Is:

[0036]

[0037] The sensitivity of the multi-point conformal design constraint to the microscopic design variable Is:

[0038]

[0039] Where Is the artificial displacement vector

[0040] Furthermore, the sensitivity analysis of the objective function in S8 is as follows:

[0041] The sensitivity of the objective function to the macroscopic design variables is:

[0042]

[0043] where By separately calculating the element stiffness matrix of the th layer and the th element of the laminate: For calculation:

[0044]

[0045]

[0046] where is the macroscopic penalty parameter, is the microscopic element elastic constitutive matrix;

[0047] The sensitivity of the objective function to the microscopic design variable is:

[0048] .

[0049] Furthermore, the convergence criterion in S9 is:

[0050]

[0051] where is the convergence criterion, is the objective function at the th step, is the objective function at the th step, is the maximum objective function in the iteration, is the minimum objective function in the iteration, is the convergence tolerance.

[0052] The beneficial effects of the present invention are:

[0053] (1) The present invention clearly gives the derivation of the volume constraint, multi-point conformal constraint, and analytical sensitivity of the objective function;

[0054] (2) The present invention solves the difficulty that there is no actual physical element for calculating the local deformation energy of the structure in the open-hole structure through the artificial weak element concept;

[0055] (3) The present invention is applicable to any open-hole structure of any fiber-reinforced composite material. Description of the Drawings

[0056] Figure 1 It is a flowchart of the variable stiffness multi - point conformal multi - scale optimization design method for fiber - reinforced composites.

[0057] Figure 2 It is a schematic diagram of the multi - point conformal design and the artificial weak element concept of the present invention.

[0058] Figure 3 It is a schematic diagram of the calculation structure of an L - shaped beam with a square hole.

[0059] Figure 4 It is a schematic diagram of the optimization design result of the variable stiffness multi - scale of ordinary fiber - reinforced composites.

[0060] Figure 5 It is a schematic diagram of the optimization design result of the variable stiffness multi - point conformal multi - scale of fiber - reinforced composites of the present invention. Detailed Embodiments

[0061] The present invention will be further described below in conjunction with the drawings and specific embodiments.

[0062] Based on the normal - distribution fiber optimization interpolation format, the present invention introduces the concepts of multi - point conformal design and artificial weak elements, conducts research on the variable stiffness multi - scale optimization design of multi - point conformal fiber - reinforced composites, 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 multi - scale optimization design method for fiber - reinforced composites.

[0063] As Figure 1 shown, a variable stiffness multi - point conformal multi - scale optimization design method for fiber - reinforced composites includes the following steps:

[0064] S1: Divide the fiber - reinforced composite structure into regions, including dividing the region to be designed into a design domain and the structure opening region into a conformal domain;

[0065] S2: Conduct finite - element cell division on the design domain and divide it into design cells;

[0066] S3: Conduct artificial weak - element division on the conformal domain and divide it into artificial weak elements;

[0067] S4: Initialize the macroscopic design variables and microscopic design variables;

[0068] S5: Based on the initialization result, calculate the volume constraint for the design domain;

[0069] S6: Calculate the multi - point conformal design constraint for the conformal domain;

[0070] S7: Perform finite element calculations on the overall design structure and obtain the objective function;

[0071] S8: Conduct volume constraint sensitivity analysis, multi-point conformal design constraint sensitivity analysis, and objective function sensitivity analysis;

[0072] S9: Based on the sensitivity analysis results, determine whether the convergence criterion is met. If so, complete the variable stiffness multi-point conformal multi-scale optimization design of fiber-reinforced composites; otherwise, return to step S5.

[0073] When performing conformal design on the hole area, there are no physical elements for calculating the stiffness matrix of the conformal area Therefore, the present invention introduces the concepts of multi-point conformal design and artificial weak elements. As Figure 2 shown, its main idea is to consider multiple key points on the hole edge that need to control deformation and establish artificial weak elements.

[0074] In the above-mentioned S5, the volume constraint calculation of the design domain is carried out according to the formula:

[0075]

[0076]

[0077]

[0078] Where is the volume of the design domain , is the initially given volume fraction, is the sum of the volumes of the design domain , is the volume of the th design element, is the macroscopic design variable, is the element area, is the laminate thickness.

[0079] The total deformation of the structural conformal domain under the action of the load is expressed as the nodal displacement vector within the region, which includes the rigid body displacement and the warping deformation . The specific expression is:

[0080]

[0081] Since the conformal design is to achieve the effect of coordinated displacement without warping deformation, the rigid body displacement of the structure is not considered in the present invention. The degree of warping deformation is quantitatively described using the regional strain energy.

[0082] To prevent numerical singularities, very small material properties are assigned to artificial weak elements so that the corresponding regional strain energy can be calculated. .

[0083] Considering that the ideal conformal design effect cannot be achieved in actual engineering problems ( ), thus a is given to limit the warping deformation degree of the opening area.

[0084] In step S6, the multi-point conformal design constraint calculation for the conformal domain is performed, and the formula is:

[0085]

[0086]

[0087]

[0088] where, is the strain energy of the conformal domain , is the set parameter, is the warping deformation of the conformal domain , is the stiffness matrix of the conformal domain , and the superscript represents the transpose of the matrix.

[0089] The overall finite element calculation of the structure is carried out. Using the first-order shear deformation theory and assuming the structure is linearly elastic, the static response of the multi-scale composite laminate is obtained. Therefore, the element stiffness array of the th element in the th layer of the laminate can be obtained by integrating the elastic constitutive matrix of the macroscopic alternative materials in the design domain .

[0090] The structural linear static equilibrium equation is expressed as:

[0091] In step S7, the finite element calculation of the overall design structure is performed, and the formula is:

[0092]

[0093] where, is the element stiffness matrix of the th layer and the th element of the laminate, is the strain-displacement matrix, is the design domain Elastic constitutive matrix of medium-scale alternative materials, is the micro design variable;

[0094] Taking the minimum structural compliance as the objective function:

[0095]

[0096]

[0097] where, is the structural compliance, is the total displacement, is the total stiffness, is the total external load vector.

[0098] Based on the multi-point conformal multi-scale variable stiffness optimization framework of fiber-reinforced composites, considering the overall material consumption and the deformation energy of the hole area as constraints, and taking the minimum structural compliance as the objective function, the mathematical formula of this optimization problem is expressed as the following formula:

[0099]

[0100]

[0101]

[0102] where, is the set of design variables, is the type of alternative fiber laying angles.

[0103] The sensitivity analysis of the volume constraint in S8 is:

[0104]

[0105] where, represents taking the partial derivative.

[0106] The sensitivity analysis of the multi-point conformal design constraint in S8 is:

[0107] The sensitivity calculation of the multi-point conformal design constraint to the macro design variable is:

[0108]

[0109] where is a symmetric matrix, , the above formula can be combined and expressed as:

[0110]

[0111] The nodal displacement of the artificial weak element It can be extracted from the overall structural displacement through the coefficient matrix The relationship between the partial derivative of the nodal displacement of the artificial weak element with respect to and the overall displacement

[0112]

[0113] The nodal displacement of the artificial weak element is established For The partial derivative with respect to and the overall displacement For The relationship of the partial derivative is as follows:

[0114]

[0115] Substituting into the structural static analysis and integrating, we get:

[0116]

[0117] Since it is difficult to directly calculate the inverse matrix of the structural stiffness matrix in the actual calculation process, the adjoint method is used for further derivation. Let:

[0118]

[0119] Taking as the artificial load vector and applying it to the overall structure, and performing finite element analysis to obtain the artificial displacement vector :

[0120]

[0121] Integrating the above derivation process, it can be expressed as:

[0122]

[0123] The sensitivity of the multi-point conformal design constraint to the macroscopic design variable is:

[0124]

[0125] The sensitivity of the multi-point conformal design constraint to the microscopic design variable is:

[0126]

[0127] Among them, is the artificial displacement vector, , are both 0.

[0128] The sensitivity analysis of the objective function in S8 is:

[0129] The objective function with respect to the macroscopic design variable The sensitivity of

[0130]

[0131] is: By separately calculating the element stiffness matrix of the th layer and the th element of the laminate: The calculation is as follows:

[0132]

[0133]

[0134] wherein, is the macroscopic penalty parameter, is the microscopic element elastic constitutive matrix;

[0135] The sensitivity of the objective function with respect to the microscopic design variable is:

[0136]

[0137] The convergence criterion in S9 is:

[0138]

[0139] wherein, is the convergence criterion, is the objective function at the th step, is the objective function at the th step, is the maximum objective function in the iteration, is the minimum objective function in the iteration, is the convergence tolerance.

[0140] In an embodiment of the present invention, as shown in Figure 3 , a L-shaped beam structure with a square hole is given and a response load is applied, as shown in Figure 4 and Figure 5As shown in the figure, by comparing the results obtained under the ordinary and the proposed solutions of 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 displacements of the 4 key points become larger, but the relative displacements become smaller, especially for 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 macroscopic configuration, after applying multi-point conformal constraints, the structure in the lower left corner is closer to being hinged. After applying multi-point conformal constraints, there are also cases where the relative displacements between key points become larger. For example, for key points 14 (0.886% in the X direction and 139.25% in the Y direction), but the absolute displacements between these two points are smaller, which is considered to sacrifice the displacement at this position during the optimization process to coordinate the relative displacements between key points 12, 23, and 34. In summary, this numerical example fully demonstrates the effectiveness of the present invention.

[0141] Those of ordinary skill in the art will realize that the embodiments described herein are for helping the reader understand the principles of the present invention, and it should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific deformations and combinations that do not depart from the essence of the present invention based on the technical revelations disclosed in the present invention, and these deformations 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; The volume constraint sensitivity analysis is: in, represents partial derivative, Design domain The volume and is the macro design variable, is the unit area, is the thickness of the laminate, is the number of laminate layers; The multi-point conformal design constraint sensitivity analysis 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, Conformal Domain The strain energy, is the artificial displacement vector, with superscript represents the transpose of a matrix, is the overall external load vector, is the overall displacement; The objective function sensitivity analysis 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, For structural flexibility, is the overall stiffness, is the strain-displacement matrix, Design domain Elastic constitutive matrix of meso-macroscopic candidate materials, is the macro penalty parameter, is the elastic constitutive matrix of the micro-element; Objective function for micro-design variables The sensitivity is: 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 1, 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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