Consider the transverse isotropic topology optimization method of short carbon fiber composites under the return path

Abstract

The present invention is applicable to the field of material structure optimization design, and provides a transverse isotropic topology optimization method for short carbon fiber composites considering the return path, including the following steps: measuring independent parameters in different directions of the transverse isotropic material; obtaining the compliance matrices of the transverse isotropic material in the direction along the printing filament and perpendicular to the filament direction; constructing a unit stiffness matrix solution model applicable to the transverse isotropic material; assembling to obtain the overall stiffness matrix; taking the minimization of the structural compliance as the objective and the relative volume fraction of the structure as the constraint, using the compliance and the relative volume fraction to iteratively solve the sensitivity of the design variables. When the current design meets the volume constraint, the absolute value difference of the design variables between two consecutive iterations is less than the preset value or the number of cycles is greater than the maximum number of cycles, the iteration ends, and the optimal structure is obtained. The present invention enables the topology optimization result to fully consider the characteristics of the fused deposition method and the filament manufacturing process, and improves the mechanical properties of the topology optimization structure.

Description

Technical Field

[0001] The present invention relates to the field of material structure optimization design, and particularly to a transverse isotropic topology optimization method for short carbon fiber composites considering the return path. Background Technique

[0002] Topology optimization refers to a design method that seeks to achieve lightweight and multifunctional innovative structures while meeting the high performance of parts by reasonably allocating various materials and determining the optimal force transmission path. Different from size optimization and shape optimization, it has a higher design freedom, and the structure is often very novel. However, the components obtained by topology optimization often have complex shapes, and it is very difficult or even impossible to process them by traditional additive methods. Therefore, it is necessary to use additive manufacturing technology to stack complex parts layer by layer. The currently relatively mature 3D printing technology is the fused deposition technology. It feeds filaments made of some thermoplastic resins into the printing nozzle, and then extrudes a certain amount of molten resin filaments. Through the layer-by-layer stacking between the molten filaments and cooling, a solid is finally formed. However, the printing accuracy of fused deposition modeling (FDM) is relatively low, and there are pores formed by the layer-by-layer stacking between the fused filaments inside the products of this 3D printing technology, resulting in the mechanical properties of the printed products being lower than those of the products processed by general traditional processes. Therefore, the strength of the products formed by FDM can no longer meet the requirements in the fields of automobiles, medical treatment, mold manufacturing, etc. In order to improve the strength of 3D printed products, it is necessary to enhance and modify the 3D printing materials. Short carbon fibers have excellent mechanical properties and are recognized as one of the best materials for resin reinforcement.

[0003] Topology optimization has become a powerful tool for structural performance optimization and is also a research hotspot in the field of structural optimization in the past decade. Its representative methods include the level set method, the ESO method (evolutionary structural optimization), the SIMP method (solid isotropic material with penalization), etc. However, traditional topology optimization is generally based on single materials or isotropic materials, while additive structures have been proven to have obvious anisotropy. It is found that during the fused deposition 3D printing process, when the molten resin is extruded from the nozzle, the short carbon fibers will generate one-dimensional orientation at the nozzle. When the molten resin is extruded from the nozzle, the speed is very low, and the orientation of the short fibers can be retained to a certain extent. At the same time, FDM is a fused filament deposition process, and the strength along the fiber direction of the fused filaments is greater than the bonding strength between the fused filaments. Therefore, the products printed by 3D printing will exhibit anisotropy.

[0004] Due to its superior material properties such as high specific strength, high specific stiffness, excellent corrosion resistance, fatigue resistance and other physical properties, fiber-reinforced composite materials have been widely used in the fields of aerospace, automotive, new energy equipment, etc. With the development of topology optimization technology and the growth of multi-functional design requirements, composite material topology optimization has attracted more and more attention. The structural form of composite materials is bound to bring higher structural performance, which also poses challenges to traditional composite material topology optimization methods.

[0005] Among the existing composite material structure optimization methods at home and abroad, they are usually based on the isotropic assumption. Most of them do not consider the anisotropy of composite materials to optimize the structure or the fiber laying angle, and cannot optimize the design for the above-mentioned multi-material, multi-scale and variable stiffness complex problems. The performance of the materials is not fully utilized, resulting in waste of materials. Compared with the double-material hybrid homogenization, composite materials with anisotropic structures have obvious performance improvement in specific directions. If the anisotropy of the additive is considered when performing topology on the structure of the composite material, the material utilization rate can be improved while maintaining the original performance, providing a design result with better performance for structure optimization and saving costs. Therefore, in order to solve the problems of anisotropic composite materials that cannot be solved by traditional topology optimization methods, it is necessary to propose a topology optimization method applicable to composite materials. That is, it is necessary to provide a transverse isotropic topology optimization method for short carbon fiber composites considering the return path to solve the above problems. Summary of the Invention

[0006] Aiming at the deficiencies of the existing technology, the purpose of the present invention is to provide a transverse isotropic topology optimization method for short carbon fiber composites considering the return path to solve the problems in the above-mentioned background technology.

[0007] The present invention is realized as follows. A transverse isotropic topology optimization method for short carbon fiber composites considering the return path, the method includes the following steps:

[0008] Step 1, print experimental specimens with short carbon fiber-reinforced composite filaments, and measure the independent parameters in different directions of the transversely isotropic material;

[0009] Step 2, initialize the design variable parameters according to the independent parameters, and obtain the compliance matrices S1 and s2 of the transversely isotropic material in the direction along the printed filament and perpendicular to the filament direction from the generalized Hooke's law;

[0010] Step 3, based on the isotropic material finite element analysis to solve the element stiffness matrix model of the eight-node regular hexahedron element, construct an element stiffness matrix solution model applicable to the transversely isotropic material;

[0011] Step 4: Based on the improved isotropic material penalty method, after interpolating the element stiffness matrix, the global stiffness matrix K is assembled according to the influence of the displacement of each node degree of freedom in each element by other nodes;

[0012] Step 5: Solve the node displacement vector U according to the formula KU = F, and establish a topological optimization model of the transversely isotropic material with the goal of minimizing the structural compliance and the structural volume as the constraint;

[0013] Step 6: Solve the sensitivity of the objective function to the design variables according to the relationship between the element stiffness matrix and the design variables and the node displacement vector U;

[0014] Step 7: Use the standard optimality criterion algorithm, with the goal of minimizing the structural compliance and the relative volume fraction of the structure as the constraint, and iteratively solve the sensitivity of the design variables using the compliance and the relative volume fraction. During the iteration process, if the current design does not meet the relative volume constraint, or the absolute value difference between the design variables of the previous and current iterations is greater than the preset value, return to Step 4 for a new round of iterative optimization; otherwise, proceed to Step 8;

[0015] Step 8: If the current design meets the volume constraint, the absolute value difference between the design variables of the previous and current iterations is less than the preset value, or the number of cycles is greater than the maximum number of cycles, the iteration ends, and the optimal structure of the transversely isotropic material topological optimization is obtained.

[0016] As a further solution of the present invention: The independent parameters are determined through compression and shear experiments, and the independent parameters include E1, E2, ν 12 , ν 23 and G 12 , where E1 is the elastic modulus in the X direction, E2 is the elastic modulus in the Y and Z directions, ν 12 is the ratio of the strain in the X direction to the strain in the Y direction, ν 23 is the ratio of the strain in the Y direction to the strain in the Z direction, and G 12 represents the shear strength of the XY plane.

[0017] As a further solution of the present invention: Along the X direction, the compliance matrix S1 of the material with isotropy on the YZ plane:

[0018]

[0019] Along the Y direction, the compliance matrix S2 of the material with isotropy on the XZ plane:

[0020]

[0021] As a further solution of the present invention: The calculation formula for the element stiffness matrix of the transversely isotropic material is:

[0022]

[0023]

[0024] Among them, and are respectively the transverse element stiffness matrix and the vertical element stiffness matrix of the eight-node hexahedron transversely isotropic element. B is the strain-displacement matrix, C1 is S1 -1 , C2 is S2 -1 , C1 and C2 are material stiffness matrices, and η1, η2, and η3 respectively represent the coordinates of local nodes in the eight-node hexahedron element.

[0025] As a further solution of the present invention: The calculation formula for the overall stiffness matrix K is:

[0026]

[0027] where E1 is the elastic modulus of the element with the main direction in the X direction, E2 is the elastic modulus of the element with the main direction in the Y direction, and p is the penalty factor.

[0028] As a further solution of the present invention: Establish a topology optimization model for transversely isotropic materials as:

[0029] find x = [x1, x2,..., x e …, x n T

[0030]

[0031]

[0032] x ∈ χ, χ = {x ∈ R n : 0 ≤ x ≤ 1}.

[0033] As a further solution of the present invention: Solve the sensitivity of the objective function to the design variables:

[0034]

[0035] where the nodal displacement vector of the transverse element, is the nodal displacement vector of the vertical element.

[0036] As a further solution of the present invention: The compliance matrix S1 is applicable to materials that are isotropic in the YZ plane along the X direction, where the X direction is the wire direction of the material, and the compliance matrix S2 is applicable to materials that are isotropic in the XZ plane along the Y direction, where the Y direction is the wire direction of the material.​

[0037] As a further solution of the present invention: The independent parameters of the transversely isotropic material are measured by experiments on 3D printed specimens of this material.

[0038] Compared with the prior art, the beneficial effects of the present invention are:

[0039] The present invention prints experimental specimens with short carbon fiber reinforced composite filaments, measures the independent parameters in different directions of the transversely isotropic material, constructs the element stiffness matrix of the transversely isotropic element, assembles the element stiffness matrix according to the direction to obtain the global stiffness matrix, and then conducts topology optimization to obtain a better configuration. Through the design of the transversely isotropic element, the topology optimization result fully considers the characteristics of the fused deposition method and the filament manufacturing process, and solves the problems that cannot be solved by topology optimization considering only single isotropic materials or combinations of multiple isotropic materials. The present invention can improve the mechanical properties of the topology optimized structure and reduce the economic cost while ensuring that the structure volume meets certain constraint conditions.

[0040] Based on the traditional isotropic material penalty method (SIMP), the present invention has been improved, especially in the part of solving the stiffness matrix of carbon fiber composite materials. A new idea is proposed, considering the directionality of anisotropic materials, making this method applicable to the topology optimization of carbon fiber composite materials, and can be used in many aspects such as aviation parts and medical devices. While ensuring high performance in specific directions with strict material requirements, it reduces or even removes materials in non-loading directions, avoiding unnecessary waste and reducing unnecessary cost expenditures of companies and enterprises. The present invention will help solve the problem of low material utilization rate in the topology optimization of composite materials based on the isotropic assumption of existing topology optimization technologies. Description of the Drawings

[0041] Figure 1 It is a flow chart of a transversely isotropic topology optimization method for short carbon fiber composites considering the return path.

[0042] Figure 2 It is a schematic diagram of the positions of the horizontal and vertical elements under the return path in a transversely isotropic topology optimization method for short carbon fiber composites considering the return path.

[0043] Figure 3 It is a schematic diagram of the local node positions of the transversely isotropic element in a transversely isotropic topology optimization method for short carbon fiber composites considering the return path.

[0044] Figure 4 It is a schematic diagram of the initial model of an embodiment in a transversely isotropic topology optimization method for short carbon fiber composites considering the return path.

[0045] Figure 5 It is an optimization result diagram obtained under different paths (X direction, Y direction, and the return path) in a transverse isotropic topology optimization method for short carbon fiber composites considering the return path. Specific implementation manners

[0046] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0047] The following describes in detail the specific implementation of the present invention with reference to specific embodiments.

[0048] As Figure 1 shown, an embodiment of the present invention provides a transverse isotropic topology optimization method for short carbon fiber composites considering the return path. The method includes the following steps:

[0049] Step 1: Print experimental specimens with short carbon fiber reinforced composite filaments, and measure independent parameters in different directions of the transverse isotropic material;

[0050] Step 2: Initialize the design variable parameters according to the independent parameters, and obtain the compliance matrices S1 and S2 of the transverse isotropic material in the direction along the printing filament and perpendicular to the filament direction from the generalized Hooke's law;

[0051] Step 3: Solve the element stiffness matrix model of the isotropic material based on the finite element analysis of the eight-node regular hexahedron element, and construct an element stiffness matrix solution model applicable to the transverse isotropic material;

[0052] Step 4: Based on the improved isotropic material penalty method, after interpolating the element stiffness matrix, assemble the overall stiffness matrix K according to the influence of the displacement of each node degree of freedom in each element by other nodes;

[0053] Step 5: Solve the node displacement vector U according to the formula KU = F, and establish a topology optimization model of the transverse isotropic material with the goal of minimizing the structural compliance and the structural volume as the constraint;

[0054] Step 6: Solve the sensitivity of the objective function to the design variables according to the relationship between the element stiffness matrix and the design variables and the node displacement vector U;

[0055] Step 7: Using the standard optimality criterion algorithm, with the goal of minimizing the structural flexibility and the relative volume fraction of the structure as the constraint, the sensitivities of the design variables with respect to flexibility and relative volume fraction are iteratively solved. During the iteration process, if the current design does not meet the relative volume constraint, or the absolute value difference between the design variables of two consecutive iterations is greater than the preset value, return to Step 4 for a new round of iterative optimization; otherwise, proceed to Step 8;

[0056] Step 8: If the current design meets the volume constraint, the absolute value difference between the design variables of two consecutive iterations is less than the preset value, or the number of loops is greater than the maximum number of loops, the iteration ends, and the optimal structure of the transverse isotropic material topology optimization is obtained.

[0057] As Figure 2 and Figure 4 shown, in the embodiment of the present invention, the independent parameters of the transverse isotropic material are measured by experiments on 3D printed specimens of the material. The independent parameters are determined through compression and shear experiments, and the independent parameters include E1, E2, ν 12 , v 23 and G 12 , where E1 is the elastic modulus in the X direction, E2 is the elastic modulus in the Y and Z directions, v 12 is the ratio of the strain in the X direction to the strain in the Y direction, v 23 is the ratio of the strain in the Y direction to the strain in the Z direction, and G 12 represents the shear strength of the XY plane.

[0058] In the embodiment of the present invention, along the X direction, the flexibility matrix S1 of the material with isotropy on the YZ plane:

[0059]

[0060] Along the Y direction, the flexibility matrix S2 of the material with isotropy on the XZ plane:

[0061]

[0062] The flexibility matrix S1 is applicable to the material with isotropy on the YZ plane along the X direction, and the X direction is the wire direction of the material; the flexibility matrix S2 is applicable to the material with isotropy on the XZ plane along the Y direction, and the Y direction is the wire direction of the material.

[0063] As Figure 3 shown, in the embodiment of the present invention, specific local node numbering and shape function construction of the unit are adopted to construct a solution model for the unit stiffness matrix of the transverse isotropic material. The local node (8 nodes) positions of the eight-node regular hexahedron unit are numbered as N1, N2, N3, N4, N5, N6, N7, and N8 respectively. The coordinates of the 8 nodes are shown in the following table:

[0064]

[0065] The shape functions are obtained from the Lagrange interpolation formula as follows:

[0066]

[0067] where e = 1, 2, 3 and q = 1,..., 8; the strain-displacement matrix B is solved by the following formula:

[0068]

[0069] The element stiffness matrix of the transversely isotropic material can be obtained from the following formula:

[0070]

[0071]

[0072] where and are the transverse element stiffness matrix and the vertical element stiffness matrix of the eight-node hexahedron transversely isotropic element respectively, B is the strain-displacement matrix, C1 is S1 -1 , C2 is S2 -1 , C1 and C2 are the material stiffness matrices, and η1, η2 and η3 represent the coordinates of the local nodes in the eight-node hexahedron element respectively. For the transversely isotropic material, C1 is:

[0073]

[0074] where

[0075]

[0076] In the embodiments of the present invention, based on the improved isotropic material penalty method (SIMP), after interpolating the element stiffness matrix, considering the influence of the displacements of each node degree of freedom of each element by other nodes, the global stiffness matrix K is assembled as follows:

[0077]

[0078] where E1 is the elastic modulus of the element with the main direction in the X direction, E2 is the elastic modulus of the element with the main direction in the Y direction, and p is the penalty factor.

[0079] In the embodiments of the present invention, the topology optimization model of the transversely isotropic material is established as follows:

[0080] find x = [x1, x2,..., x e ..., x n T ​

[0081]

[0082]

[0083] x ∈ χ, where χ = {x ∈ R n : 0 ≤ x ≤ 1}.

[0084] In the embodiments of the present invention, the sensitivity of the objective function to the design variables is solved:

[0085]

[0086] where is the nodal displacement vector of the horizontal element, is the nodal displacement vector of the vertical element.

[0087] The optimization results of the present invention under different paths of a certain parameter are as Figure 5 shown. It can be seen that the structure obtained by the present invention considers the transverse isotropy of the short carbon fiber reinforced composite material under the return path. On the premise of satisfying the imposed volume constraint condition, the minimization of the structural compliance is achieved, and the mechanical properties of the material are fully exerted. Through the design of the transverse isotropic element in the embodiments of the present invention, the topology optimization result fully considers the characteristics of the fused deposition method and the wire manufacturing process, and solves the problems that cannot be solved by topology optimization that only considers single isotropic materials or the combination of multi-isotropic materials. The embodiments of the present invention can improve the mechanical properties of the topology optimized structure and reduce the economic cost while ensuring that the structure volume meets certain constraint conditions.

[0088] The above only describes the preferred embodiments of the present invention in detail and is not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

[0089] It should be understood that although the steps in the flowcharts of the embodiments of the present invention are shown in sequence according to the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless there is a clear indication in this article, the execution of these steps has no strict order limit, and these steps can be executed in other orders. Moreover, at least a part of the steps in each embodiment may include multiple sub-steps or multiple stages. These sub-steps or stages are not necessarily executed at the same time, but can be executed at different times. The execution order of these sub-steps or stages is not necessarily sequential, but can be executed alternately or alternately with at least a part of other steps or sub-steps or stages of other steps.

[0090] Other embodiments of the present disclosure will be readily apparent to those skilled in the art after considering the disclosure in the specification and the embodiments. This application is intended to cover any variations, uses, or adaptations of the present disclosure that follow the general principles of the present disclosure and include known common general knowledge or conventional technical means in the technical field not disclosed in the present disclosure. The specification and the embodiments are only regarded as exemplary, and the true scope and spirit of the present disclosure are pointed out by the claims.

Claims

1. Consider the transverse isotropic topology optimization method of short carbon fiber composites under the return path, characterized in that, The method includes the following steps: Step 1: Print experimental specimens with short carbon fiber reinforced composite filaments, and measure the independent parameters in different directions of the transversely isotropic material; Step 2: Initialize the design variable parameters according to the independent parameters. From the generalized Hooke's law, obtain the compliance matrices S1 and S2 of the transversely isotropic material in the direction along the printing filament and the direction perpendicular to the filament; Step 3: Based on the finite element analysis of the isotropic material of the eight-node regular hexahedron element to solve the element stiffness matrix model, and construct an element stiffness matrix solution model applicable to the transversely isotropic material; Step 4: Based on the improved penalty method for isotropic materials, after interpolating the element stiffness matrix, according to the influence of the displacement of each node degree of freedom in each element by other nodes, assemble to obtain the global stiffness matrix K; Step 5: Solve to obtain the node displacement vector U according to the formula KU = F. Taking the minimization of the structural compliance as the objective and the structural volume as the constraint, establish a topology optimization model for the transversely isotropic material; Step 6: According to the relationship between the element stiffness matrix and the design variables and the node displacement vector U, solve the sensitivity of the objective function to the design variables; Step 7: Use the standard optimality criterion algorithm, taking the minimization of the structural compliance as the objective and the relative volume fraction of the structure as the constraint, and use the compliance and relative volume fraction to iteratively solve the sensitivity of the design variables. During the iteration process, if the current design does not meet the relative volume constraint, or the absolute value difference of the design variables between two consecutive iterations is greater than the preset value, return to Step 4 for a new round of iterative optimization, otherwise, proceed to Step 8; Step 8: If the current design satisfies the volume constraint, and the difference between the absolute values ​​of the design variables of the previous and next iterations is less than the preset value or the number of cycles is greater than the maximum number of cycles, the iteration ends and the optimal structure of the transversely isotropic material topology optimization is obtained. The independent parameters are determined by compression and shear tests, and the independent parameters include E1, E2, v 12 、v 23 and G 12 , E1 is the elastic modulus in the X direction, E2 is the elastic modulus in the Y and Z directions, v 12 is the ratio of the strain in the X direction to the strain in the Y direction, v 23 is the ratio of strain in the Y direction to strain in the Z direction, G 12 represents the shear strength of the XY plane, and the calculation formula of the overall stiffness matrix K is: Among them, E1 is the elastic modulus of the unit with the main direction in the X direction, E is the elastic modulus of the unit with the main direction in the Y direction, p is the penalty factor, and the topology optimization model of the transversely isotropic material is established as follows: find x=[x1,x2,…,x e …,x n T ​ x ∈ χ, where χ = {x ∈ R n : 0 ≤ x ≤ 1}, find the sensitivity of the objective function with respect to the design variables: Among them, the nodal displacement vector of the horizontal element, is the nodal displacement vector of the vertical element.

2. The transverse isotropic topology optimization method of short carbon fiber composites under the return path according to claim 1, characterized in that, In the X direction, the compliance matrix S1 of the material with isotropy in the YZ plane: In the Y direction, the compliance matrix S2 of the material with isotropy in the XZ plane:

3. The transverse isotropic topology optimization method of short carbon fiber composites under the return path according to claim 1, characterized in that, The calculation formula for the element stiffness matrix of the transversely isotropic material is: Among them, and are respectively the transverse element stiffness matrix and the vertical element of the eight-node hexahedron transversely isotropic element, B is the strain-displacement matrix, C1 is S1 -1 , C2 is S2 -1 , C1 and C2 are material stiffness matrices, and η1, η2 and η3 respectively represent the coordinates of the local nodes in the eight-node hexahedron element.

4. The transverse isotropic topology optimization method of short carbon fiber composites under the return path according to claim 1, characterized in that, The compliance matrix S1 is applicable to the material with isotropy in the YZ plane in the X direction, where the X direction is the filament direction of the material, and the compliance matrix S2 is applicable to the material with isotropy in the XZ plane in the Y direction, where the Y direction is the filament direction of the material.

5. The transverse isotropic topology optimization method of short carbon fiber composites under the return path according to claim 1, characterized in that, The independent parameters of the transversely isotropic material are measured by experiments on 3D printed specimens of the material.

Images