Topological optimization method for additive and subtractive composite manufacturing

Abstract

The invention relates to a topological optimization method for additive and subtractive composite manufacturing. The method comprises the steps that design variables are mapped into a structure density function and an auxiliary support density function through density filtering and a step function; a convection-diffusion equation is solved based on a structural density function to obtain a solution, a solution with a clear boundary is obtained through step function mapping, then machinable auxiliary support performance evaluation parameters are calculated, a virtual heat conduction problem is constructed based on the evaluation parameters, the maximum temperature is calculated, and structural rigidity or flexibility is evaluated by solving a linear elasticity equation. And iteratively optimizing the design variables by using a moving asymptote method until the algorithm converges, and finally outputting an optimized structure density function and an auxiliary support performance evaluation parameter. According to the method, by optimizing the design variables of the structure and the auxiliary support, the machining quality and efficiency of additive and subtractive composite manufacturing are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of additive manufacturing, and in particular to a topology optimization method for additive and subtractive composite manufacturing. Background Art

[0002] As an advanced structural design method, topology optimization can find the optimal material distribution form in the design area to improve the structural performance. Additive manufacturing (also known as 3D printing) is an advanced manufacturing technology that processes parts by stacking materials layer by layer on the printing plane without being restricted by geometric form. Combining topology optimization and additive manufacturing can be used to integrate the design and processing of complex industrial products. This technology has been widely used in the design and manufacture of various structures in the fields of aviation, machinery, and energy, and has important research significance and industrial application value.

[0003] Auxiliary supports are indispensable in additive manufacturing of complex structures. Auxiliary supports can not only be used to fix the overhanging surface to complete printing, but also help improve the heat dissipation efficiency in additive manufacturing, thereby improving printing quality. However, the use of auxiliary supports will lead to a significant increase in processing costs. The printing of auxiliary supports will waste processing time and materials, and the auxiliary supports placed inside the structure after processing are difficult to remove. In recent years, researchers have considered the control of auxiliary support dosage and the coordinated optimization of mechanical / heat dissipation performance in the process of structural topology optimization design. Such methods usually use simplified mechanical or heat dissipation models to simulate the mechanism of auxiliary supports in the additive manufacturing process, so as to design the structural form of auxiliary supports to improve stiffness or heat dissipation efficiency. The current auxiliary support optimization does not consider its machinability, which makes it difficult to remove the auxiliary supports arranged on the overhanging surface of the internal hole of the structure by machining. Therefore, it is urgent to consider the coordinated optimization and machinability of auxiliary supports in the structural design stage, so as to improve the performance of auxiliary supports and ensure that the optimized auxiliary supports can be removed by machining. Summary of the invention

[0004] In order to solve the above problems, the present invention proposes a topology optimization method for additive and subtractive composite manufacturing, which can support the consideration of additive and subtractive process constraints in the structural topology optimization design stage, so that the internal overhanging surfaces of the optimized structure do not require auxiliary supports, and the auxiliary supports added to the external overhanging surfaces have excellent heat dissipation efficiency and can be removed by machining, thereby improving the additive manufacturing processability and processing quality of the optimized structure.

[0005] The specific plan is as follows:

[0006] A topology optimization method for additive and subtractive composite manufacturing, comprising:

[0007] S1, through density filtering and step function, the optimized structure topology configuration design variable γ and the auxiliary support configuration corresponding design variable ρ are mapped into the structural density function and auxiliary supports in a given printing direction

[0008] S2, based on the structure density function Solve the convection-diffusion equation and obtain the solution ψ of the convection-diffusion equation;

[0009] S3, mapping the solution ψ of the convection-diffusion equation through a step function to obtain a solution with clear boundaries Based on solution and auxiliary supports in a given printing direction Calculate the machinable auxiliary support performance evaluation parameter ρ ψ ;

[0010] S4, based on auxiliary support performance evaluation parameter ρ ψ , construct a virtual heat conduction problem to evaluate the auxiliary support performance;

[0011] S5, in the virtual heat conduction problem, the maximum temperature T is obtained by constructing and calculating the maximum temperature constraint max ;

[0012] S6, constructing a topology optimization problem for maximizing structural stiffness or minimizing compliance, defining an optimization problem formula based on material volume constraints and maximum temperature constraints; obtaining the stiffness or compliance of the structure by solving a linear elastic equation, performing a sensitivity analysis on the stiffness or compliance of the structure, and obtaining the gradient of the objective function and the constraint; the objective function is compliance, and the gradient of the constraint is the gradient of the volume constraint and the maximum temperature constraint;

[0013] S7, solving the optimization problem by the moving asymptote method, and updating the design variable γ of the optimized structure topology configuration and the corresponding design variable ρ of the auxiliary support configuration;

[0014] S8, determine whether the optimization algorithm converges; if not, re-execute S1-S7; if converged, output the structure density function and auxiliary support ρ under a given printing direction ψ .

[0015] Furthermore, in S2, the convection-diffusion equation is calculated as follows:

[0016]

[0017] Where Ω represents the design area, Γ is the boundary of the design area, is the boundary where light directly shines, is the boundary where light cannot directly illuminate, L represents the projection length of the design area in the processing direction, D represents the diffusion tensor, v represents the advection vector, β ψ represents the magnitude of the source term.

[0018] Further, in S3, the machinable auxiliary support performance evaluation parameter ρ is calculated ψ , the formula is as follows:

[0019]

[0020] Furthermore, in S4, the equilibrium equation of the virtual heat conduction is:

[0021]

[0022] Among them, T 0 For a fixed temperature, is the boundary defining the fixed temperature; q t represents the heat flow, Represents the structure / auxiliary support interface; It represents the thermal conductivity, and the calculation formula is as follows:

[0023]

[0024] in, and k min represent the thermal conductivity of the structural entity material, auxiliary support entity material and the empty phase respectively; T p and T s represents the anisotropic control tensor of the structure and auxiliary support; let ρ ψ Substituting the equations (1) and (2) To evaluate the auxiliary support performance.

[0025] Furthermore, in S5, the calculation formula of the maximum temperature constraint is as follows:

[0026]

[0027] Among them, q p represents the penalty factor, q p =16; represents the reference temperature; ξ represents the user-defined coefficient.

[0028] Furthermore, in S6, the optimization problem formula is as follows:

[0029]

[0030] Where J is the flexibility of the structure, and F are the stiffness matrix and force matrix of the linear elastic problem, respectively. K ψand F ψ They represent the stiffness matrix and load matrix of the convection-diffusion equation, respectively, and K T and F T They represent the stiffness matrix and load matrix of the virtual heat conduction equation respectively. and Respectively represent the material volume usage of structure and auxiliary support.

[0031] The present invention adopts the above technical solution and has the following beneficial effects:

[0032] (1) The present invention optimizes the design variables of the structure and auxiliary supports to ensure that the auxiliary supports can be removed by machining after printing is completed, thereby improving the processing quality and efficiency of additive manufacturing;

[0033] (2) The present invention optimizes the layout of the auxiliary support through a virtual heat conduction equation, so that it has excellent heat dissipation performance, thereby improving the heat dissipation effect during the printing process and improving the printing quality;

[0034] (3) The optimized structure of the present invention does not require auxiliary supports on the internal overhanging surface, which reduces the use of auxiliary supports, reduces the waste of materials and processing time, and thus effectively reduces the overall processing cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 A collaborative optimization process for the structure, auxiliary support and machining direction of an embodiment of the present invention;

[0036] Figure 2 Optimizing virtual heat conduction problem settings for auxiliary supports of embodiments of the present invention;

[0037] Figure 3 This is an example of auxiliary support obtained by optimizing the virtual heat conduction equation according to an embodiment of the present invention;

[0038] Figure 4 A convection-diffusion problem setting for identification of a processable region according to an embodiment of the present invention;

[0039] Figure 5 An example of identifying machinable auxiliary supports by solving a convection-diffusion problem according to an embodiment of the present invention;

[0040] Figure 6 The collaborative optimization results of the two-dimensional bridge deck structure and auxiliary support in a given machining direction of an example of the present invention;

[0041] Figure 7 The collaborative optimization results of the two-dimensional bridge deck structure, auxiliary support and machining direction of the embodiment of the present invention;

[0042] Figure 8The three-dimensional cantilever beam problem setting of an embodiment of the present invention and the optimization result without considering the process constraints;

[0043] Fig. 9 This is the optimization result of the three-dimensional cantilever beam problem with additive and subtractive material process constraints according to an embodiment of the present invention. DETAILED DESCRIPTION

[0044] The present invention is further described in detail below in conjunction with embodiments and drawings, but the embodiments of the present invention are not limited thereto.

[0045] like Figure 1 As shown, the present invention provides a topology optimization method for additive and subtractive composite manufacturing, comprising:

[0046] S1, through density filtering and step function, the optimized structure topology configuration design variable γ and the auxiliary support configuration corresponding design variable ρ are mapped into the structural density function and auxiliary supports in a given printing direction

[0047] Specifically, Figure 2 (a) is shown by The topological structure represented by For the empty phase material, for physical materials), Represents the auxiliary supports under a given printing direction b; it can be seen that the auxiliary supports are added on the non-self-supporting overhanging surfaces of the structure (dashed lines).

[0048] S2, based on the structure density function Solve the convection-diffusion equation and obtain the solution ψ of the convection-diffusion equation;

[0049] The convection-diffusion equation calculation formula is as follows:

[0050]

[0051] Where Ω represents the design area, Γ is the boundary of the design area, is the boundary where light directly shines, is the boundary where light cannot directly illuminate, L represents the projection length of the design area in the processing direction, D represents the diffusion tensor, v represents the advection vector, β ψ represents the magnitude of the source term.

[0052] The topologically optimized structure with self-supporting internal holes still requires external auxiliary supports to be successfully printed during the additive manufacturing process. However, the lack of structural boundary display description during the topological optimization process of the structure and auxiliary supports makes it difficult to identify removable external auxiliary supports. In this embodiment, a machinable auxiliary support identification method based on the convection-diffusion equation is proposed. For a given tool processing direction and structure, the definition is as follows: Figure 4 The boundary conditions shown are identified by solving the convection-diffusion equation to identify the non-machinable area at the boundary visible to light or machining tools. Define ψ=0 at the boundary, and define The Newman boundary condition is applied at ; for a given machining direction m, the advection vector is specified as in is the size of the advection term.

[0053] S3, mapping the solution ψ of the convection-diffusion equation through a step function to obtain a solution with clear boundaries Based on solution and auxiliary supports in a given printing direction Calculate the machinable auxiliary support performance evaluation parameter ρ ψ ;

[0054] Specifically, the calculation of the machinable auxiliary support performance evaluation parameter ρ ψ , the formula is as follows:

[0055]

[0056] Specifically, for a given processing direction, the convection-diffusion equation can simulate the transmission of light and identify the shadow area as the unprocessable area; e.g. Figure 5 As shown, by solving the convection-diffusion equation, we obtain Figure 2 Structure in The machinable area for a given machining direction The right and bottom boundaries are fixed to ψ=0( Figure 4 ); Figure 5 (a) shows the solution ψ obtained by solving the convection. It can be seen that the boundary of ψ is fuzzy. In this embodiment, a step function mapping is used to obtain a clear boundary. The mapped solution is Indicates that Figure 5 As shown in (b), it can be used The machinable area was successfully identified.

[0057] S4, based on auxiliary support performance evaluation parameter ρ ψ , a virtual heat conduction problem is constructed to evaluate the auxiliary support performance.

[0058] Specifically, the balance equation of the virtual heat conduction is:

[0059]

[0060] Among them, T 0 For a fixed temperature, is the boundary defining the fixed temperature; q t represents the heat flow, represents the structure / support interface, where the boundary is imposed for heat flow; It represents the thermal conductivity, and the calculation formula is as follows:

[0061]

[0062] in, and k min represent the thermal conductivity of the structural entity material, auxiliary support entity material and the empty phase respectively; T p and T s represents the anisotropic control tensor of the structure and auxiliary support; let ρ ψ Substituting the equations (1) and (2) To evaluate the auxiliary support performance.

[0063] Specifically, the virtual heat conduction problem used to assist in heat dissipation simulation and optimization is defined as follows: Figure 2 (b) shows that in the model, on the non-self-supporting overhanging surface or the structure / auxiliary support interface Applied heat flux q t , and set the bottom printing plane temperature to T 0 , as a heat sink, the input heat flux is transferred to the heat sink through the structure and auxiliary support. In the optimization process, it is impossible to express it explicitly using curves, etc., so the present invention uses a structural density function The gradient is applied.

[0064] Specifically, Figure 5 (c) and (d) show the machinable area indicator function. The mapped machinable auxiliary supports; and Figure 3 Compared with (c), the supports in the hole and the upper left corner are removed because they are not machinable. Since the non-machinable auxiliary supports disappear on the overhanging surface and cannot be used for heat dissipation, Figure 5 The maximum temperature ratio in (e) Figure 3 The maximum temperature in is about 5 times higher; therefore, by controlling the maximum temperature in the virtual heat conduction problem, the overhanging surface of the non-machinable area can be made self-supporting.

[0065] S5, in the virtual heat conduction problem, the maximum temperature T is obtained by constructing and calculating the maximum temperature constraint max .

[0066] Specifically, the calculation formula of the maximum temperature constraint is as follows:

[0067]

[0068] Among them, q p represents the penalty factor, q p =16; represents the reference temperature; ξ represents the user-defined coefficient.

[0069] In the virtual heat conduction problem, by limiting the maximum temperature, the optimal layout of auxiliary supports can be found to improve the heat dissipation efficiency, or the use of auxiliary supports can be reduced through the design of self-supporting overhanging surfaces; Figure 3 The given design in (a) The optimized auxiliary support is as follows Figure 3 (b) shows that the corresponding temperature field distribution is Figure 3 As shown in (c), in order to transfer the heat input from the structure / support interface to the printing plane, the optimized auxiliary supports are arranged on the non-self-supporting overhanging surface.

[0070] Specifically, if the auxiliary support has good heat dissipation performance, the maximum temperature value is smaller, so the maximum temperature can reflect the support performance.

[0071] S6, construct a topology optimization problem to maximize the structural stiffness or minimize the flexibility, define the optimization problem formula based on the material volume constraint and the maximum temperature constraint; obtain the stiffness or flexibility of the structure by solving the linear elastic equation, perform sensitivity analysis on the stiffness or flexibility of the structure, and obtain the gradient of the objective function and the constraint; the objective function is the flexibility, and the gradient of the constraint is the gradient of the volume constraint and the maximum temperature constraint.

[0072] Specifically, the optimization problem formula is as follows:

[0073]

[0074] Where J is the flexibility of the structure, and F are the stiffness matrix and force matrix of the linear elastic problem, respectively. K ψ and F ψ They represent the stiffness matrix and load matrix of the convection-diffusion equation, respectively, and K T and F T They represent the stiffness matrix and load matrix of the virtual heat conduction equation respectively. and Respectively represent the material volume usage of structure and auxiliary support.

[0075] S7, solving the optimization problem by the moving asymptote method, and updating the design variable γ of the optimized structure topology configuration and the corresponding design variable ρ of the auxiliary support configuration;

[0076] S8, determine whether the optimization algorithm converges; if not, re-execute S1-S7; if converged, output the structure density function and auxiliary support ρ under a given printing direction ψ .

[0077] Specifically, the effectiveness of the proposed technology is verified through two-dimensional and three-dimensional examples in this embodiment. Figure 6 In the two-dimensional bridge deck optimization in Figure 1, for different machining directions m, the proposed topology optimization technology can find the optimal structural form and auxiliary support; for the optimized structure, the surface inclination of the internal hole is greater than the critical value of 45 degrees at the position where the tool cannot process, so no auxiliary support is required; the optimized auxiliary support is arranged on the overhanging surface of the machinable area, and the auxiliary support presents a tree-like structure, which is conducive to heat dissipation; at the same time, it can be seen that for unreasonable machining directions m = (1, 0), the structure itself needs to be greatly adjusted to meet the process constraints (such as Figure 6 (a)); in this case, the performance (stiffness) of the structure will be greatly lost, and it is necessary to further optimize the machining direction to increase the design freedom.

[0078] Specifically, Figure 7 The optimization results of collaborative optimization of machining directions are shown in Figure 2. It can be seen that for different initial machining directions, the optimization algorithm can find the machining direction that maximizes the machining area, thereby arranging auxiliary supports in a larger design area and increasing the design freedom.

[0079] Specifically, the present invention illustrates the effectiveness of the technology through the design of three-dimensional structures, such as Figure 8 The cantilever beam in (a) is the optimized structure obtained without considering the process constraints (such as Figure 8 (b) contains non-self-supporting internal holes, which require auxiliary supports during the additive manufacturing process, but it is difficult to remove the auxiliary supports after the processing is completed. The topology optimization results considering the constraints of the additive and subtractive processes are shown in Figure 2. Fig. 9 As shown, Fig. 9 (a) represents the optimized structure. Fig. 9 (b) represents the optimized auxiliary support. Fig. 9 (c) shows the optimized structure combined with the auxiliary support. It can be seen that by considering the additive and subtractive material constraints proposed in this project during the optimization process, the optimized structure achieves self-support in the non-machinable area, that is, no auxiliary support is required; while in the machinable area, the auxiliary support is arranged on the overhanging surface in the optimal form, which not only enhances the heat dissipation efficiency of the auxiliary support, but also improves the additive manufacturing quality of the optimized structure.

[0080] Finally, the optimization results show that the proposed technology can effectively find the optimal structure and auxiliary support topology and machining direction during the topology optimization process. This process not only maximizes the structural stiffness, but also optimizes the auxiliary support to improve the heat dissipation efficiency. At the same time, it ensures that the auxiliary support can be easily removed by machining, thereby reducing the overall processing cost and improving the processing quality. These advantages will help further promote the widespread application of topology optimization in the lightweight and high-performance design of future advanced equipment.

[0081] Although the present invention has been specifically shown and described in conjunction with the preferred embodiments, it should be understood by those skilled in the art that various changes may be made to the present invention in form and details without departing from the spirit and scope of the present invention as defined by the appended claims, all of which are within the scope of protection of the present invention.

Claims

1. A topology optimization method for additive and subtractive composite manufacturing, characterized in that: include: S1, through density filtering and step function, the optimized structure topology configuration design variable γ and the auxiliary support configuration corresponding design variable ρ are mapped into the structural density function and auxiliary supports in a given printing direction S2, based on the structure density function Solve the convection-diffusion equation and obtain the solution ψ of the convection-diffusion equation; S3, mapping the solution ψ of the convection-diffusion equation through a step function to obtain a solution with clear boundaries Based on solution and auxiliary supports in a given printing direction Calculate the performance evaluation parameter pψ of the machinable auxiliary support; S4, based on auxiliary support performance evaluation parameter ρ ψ , construct a virtual heat conduction problem to evaluate the auxiliary support performance; S5, in the virtual heat conduction problem, the maximum temperature T is obtained by constructing and calculating the maximum temperature constraint max ; S6, constructing a topology optimization problem for maximizing structural stiffness or minimizing compliance, defining an optimization problem formula based on material volume constraints and maximum temperature constraints; obtaining the stiffness or compliance of the structure by solving a linear elastic equation, performing a sensitivity analysis on the stiffness or compliance of the structure, and obtaining the gradient of the objective function and the constraint; the objective function is compliance, and the gradient of the constraint is the gradient of the volume constraint and the maximum temperature constraint; S7, solving the optimization problem by the moving asymptote method, and updating the design variable γ of the optimized structure topology configuration and the corresponding design variable ρ of the auxiliary support configuration; S8, determine whether the optimization algorithm converges; if not, re-execute S1-S7; if converged, output the structure density function and auxiliary support ρ under a given printing direction ψ .

2. The topology optimization method for additive and subtractive composite manufacturing according to claim 1, characterized in that: In S2, the convection-diffusion equation is calculated as follows: Where Ω represents the design area, Γ is the boundary of the design area, is the boundary where light directly shines, is the boundary where light cannot directly illuminate, L represents the projection length of the design area in the processing direction, D represents the diffusion tensor, v represents the advection vector, β ψ represents the magnitude of the source term.

3. The topology optimization method for additive and subtractive composite manufacturing according to claim 2 is characterized in that: In S3, the performance evaluation parameter ρ of the machinable auxiliary support is calculated. ψ , the formula is as follows:

4. The topology optimization method for additive and subtractive composite manufacturing according to claim 3 is characterized in that: In S4, the equilibrium equation of the virtual heat conduction is: Where T0 is a fixed temperature, is the boundary defining the fixed temperature; qt represents the heat flow, Represents the structure / auxiliary support interface; It represents the thermal conductivity, and the calculation formula is as follows: in, and k min represent the thermal conductivity of the structural entity material, auxiliary support entity material and the empty phase respectively; Tp and T s represents the anisotropic control tensor of the structure and auxiliary support; let ρ ψ Substituting the equations (1) and (2) To evaluate the auxiliary support performance.

5. The topology optimization method for additive and subtractive composite manufacturing according to claim 4, characterized in that: In S5, the calculation formula of the maximum temperature constraint is as follows: Among them, q p represents the penalty factor, q p =16; represents the reference temperature; ξ represents the user-defined coefficient.

6. The topology optimization method for additive and subtractive composite manufacturing according to claim 5, characterized in that: In S6, the optimization problem formula is as follows: min γ，ρ，θ :JC(γ,u) Where J is the flexibility of the structure, and F are the stiffness matrix and force matrix of the linear elastic problem, respectively. K ψ and F ψ They represent the stiffness matrix and load matrix of the convection-diffusion equation, respectively, and K T and F T They represent the stiffness matrix and load matrix of the virtual heat conduction equation respectively. and Respectively represent the material volume usage of structure and auxiliary support.

Images