Topology optimization method considering different overhang angle constraints and printing directions for additive manufacturing

Abstract

A topology optimization method that considers different overhang angle constraints and printing directions in additive manufacturing. It includes defining parameters such as the critical value of the structural overhang angle and establishing an optimization model; linear density filtering; density filtering using a volume conservation type Heaviside function; finite element, structural flexibility, volume and constraint value analysis; sensitivity analysis, triangle directivity sensitivity filtering; optimizing and solving the optimization model using an optimization algorithm; outputting the topology optimization results and providing the printing direction. Based on the framework of the variable density method, the present invention constructs a visible and continuous structural overhang angle constraint condition, which can be adaptively activated and tightened, and the optimization process is more stable. The design area is divided into a network using rectangular units, and the unit aspect ratio can be adjusted, so that it can be applied to different structural overhang angle critical value situations. It is applicable to different structural overhang angle constraint situations, and combined with the printing direction design, it can significantly reduce the performance loss caused by the structural overhang angle constraint.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural optimization design, and specifically relates to a topology optimization method considering different overhang angle constraints and printing directions in additive manufacturing. Background Art

[0002] Additive manufacturing is an advanced manufacturing technology that has developed rapidly in recent years. Unlike traditional subtractive manufacturing processes, additive manufacturing starts from scratch and creates parts by accumulating materials layer by layer. Additive manufacturing can form parts with complex shapes, and the more complex the part, the more significant its manufacturing speed and cost advantages. After more than 30 years of development, the quality and process of metal additive manufacturing have been fully developed. Currently, the structural parts that can be formed range from micro-nano components to large structures several meters in length, and the application areas have gradually developed from functional components to primary load-bearing structural parts. Despite this, additive manufacturing is still a relatively time-consuming and costly technology. Only by fully carrying out the design of the structure for additive manufacturing can the advantages of additive free manufacturing be realized.

[0003] Topology optimization, using numerical optimization techniques, can find the optimal material layout within certain constraints, making it an ideal combination for additive manufacturing. Compared to traditional size and shape optimization, topology optimization significantly expands the design space, enabling the design of innovative, high-performance structural layouts. The integration of topology optimization and additive manufacturing is an inevitable trend, and the industry urgently needs to develop design capabilities for additive manufacturing.

[0004] Additive manufacturing does not allow for completely free component design, and there are some unique manufacturing constraints. The structural overhang angle constraint is one of the main process constraints of additive manufacturing. It requires that the inclination angle of the lower surface of the suspended structure cannot be too large, otherwise it will collapse and warp due to lack of support and heat dissipation during the layer-by-layer manufacturing process. Research on topological optimization methods that can achieve self-supporting structures will help achieve rapid preparation and save material and time costs. The study found that the critical value of the structural overhang angle (COA) is related to the additive manufacturing powder material and process, and changing the printing direction has a great influence on the suspension characteristics of the structure.

[0005] To address the above issues, there is an urgent need to develop a topology optimization method that takes into account different overhang angle constraints and printing directions of additive manufacturing, so that topology optimization can achieve structural self-support at a lower performance cost, thereby improving structural performance, saving material costs, and shortening the design cycle. Summary of the Invention

[0006] In order to solve the above problems, the purpose of the present invention is to provide a topology optimization method that takes into account different overhang angle constraints and printing directions of additive manufacturing.

[0007] To achieve the above-mentioned object, the present invention provides a topology optimization method for additive manufacturing considering different overhang angle constraints and printing directions, comprising the following steps performed in sequence:

[0008] The topology optimization method provided by the present invention for additive manufacturing considering different overhang angle constraints and printing directions includes the following steps performed in sequence:

[0009] 1) Define the geometry of the design area, critical value of the structural overhang angle, load, and boundary conditions. Based on the initial printing direction and critical value of the structural overhang angle, mesh the design area using rectangular elements. Then, based on the SIMP framework, define the design variables, optimization objectives, and constraints, and establish an optimization model. The element density, material properties, structural material volume constraints, structural overhang angle constraint function, structural overhang angle constraint value, and optimization algorithm parameters are set. The initial value of the structural overhang angle constraint is set large to keep it in an inactive state.

[0010] 2) Performing linear density filtering on the cell density in the above optimization model to obtain the cell intermediate density;

[0011] 3) Using the volume conservation type Heaviside function to filter the above unit intermediate density to obtain the unit physical density;

[0012] 4) Based on the above-mentioned unit physical density, finite element analysis, volume constraint and structural overhang angle constraint response analysis are performed to obtain the function values of structural flexibility, total volume of structural materials and structural overhang angle constraint function;

[0013] 5) Perform sensitivity analysis on the function values of the above-mentioned structural flexibility, total volume of structural materials, and structural overhang angle constraint functions, and use a triangle directivity sensitivity filter operator to filter the sensitivity of the structural overhang angle constraint;

[0014] 6) Use the optimization algorithm to optimize and solve the optimization model. If the structural overhang angle constraint is not activated and the structural flexibility tends to be stable, set the optimal printing direction to the direction with the minimum function value of the structural overhang angle constraint function, then re-divide the mesh as needed and activate the structural overhang angle constraint; otherwise, go directly to step 7);

[0015] 7) Update the unit density and the structural overhang angle constraint value, and determine whether it has converged. If not, return to step 2) to iterate the algorithm. If it has converged, the optimization ends, and the topology optimization result is finally output, and the printing direction is given.

[0016] In step 1), the geometric shape of the design area, the critical value of the structural overhang angle (COA), the load and the boundary conditions are defined, the design area is meshed using rectangular elements according to the initial printing direction and the critical value of the structural overhang angle, and then the design variables, optimization objectives and constraints are defined based on the SIMP method framework to establish an optimization model. The element density, material properties, structural material volume constraints, structural overhang angle constraint function, structural overhang angle constraint value, and optimization algorithm parameters are set, and the method of making the initial value of the structural overhang angle constraint value large so that it is in an inactive state is as follows:

[0017] The geometric shape of the design area is defined, and the critical value of the structure's overhang angle is set to the critical value of the overhang angle of the lower surface of the structure. The design area is meshed using rectangular elements, and the aspect ratio of the elements is adjusted according to the initial printing direction and the critical value of the structure's overhang angle, so that the inclination angle of the element diagonal relative to the substrate is the same as the critical value of the structure's overhang angle. Then, based on the SIMP method framework, with minimizing structural flexibility as the optimization goal, the discretized element density and printing direction are used as design variables, and the function values of the structural material volume constraint and the structural overhang angle constraint function are used as constraints to establish an optimization model. The mathematical expression of the optimization model is:

[0018]

[0019] Where c is the structural flexibility;

[0020] d is the printing direction;

[0021] ρ is the cell density;

[0022] is the physical density of the unit after filtering;

[0023] U and K are the overall nodal displacement vector and overall stiffness matrix, respectively;

[0024] u e and k0 are the element displacement vector and element stiffness matrix respectively;

[0025] F is the external load vector of the structure;

[0026] N is the number of units;

[0027] V and V0 are the total volume of structural materials and the total volume of the design area, respectively;

[0028] V f is the volume constraint value of the structural material;

[0029] is the structural overhang angle constraint function;

[0030] ε is the structural overhang angle constraint value;

[0031] A e is the unit area;

[0032] is the density penalty function, Penalize the intermediate density, where p is the penalty factor, E min is the minimum elastic modulus, E0 is the unit elastic modulus;

[0033] In order to make the optimization process more stable, the initial value of the structural overhang angle constraint ε is made larger, so that the structural overhang angle constraint is always an invalid constraint in the early optimization process.

[0034] In step 2), the method of performing linear density filtering on the unit density in the optimized model to obtain the unit intermediate density is:

[0035] The unit intermediate density ρ * It is obtained by weighted average of the density of all cells within the circular range of the filter radius R. The specific formula is as follows:

[0036]

[0037] Among them, N e The unit set within the circular range with a filtering radius of R is centered on the unit e, H ei is the weight coefficient between unit i and unit e, and Δ(e,i) is the distance between unit i and unit e.

[0038] In step 3), the method of filtering the unit intermediate density using the volume conservation type Heaviside function to obtain the unit physical density is:

[0039] The volume conservation Heaviside function is used to calculate the cell intermediate density ρ * Filter to obtain unit physical density The specific formula is as follows:

[0040]

[0041] Among them, β is the control parameter; when the control parameter β is infinite, the cell intermediate density When the unit intermediate density is less than the parameter η 0; unit intermediate density When the unit intermediate density is greater than the parameter η is 1; the value of the parameter η is determined by bisection in each iteration to ensure that the volume of the structure remains unchanged before and after filtering.

[0042] In step 4), the method for performing finite element analysis, volume constraint and structural overhang angle constraint response analysis based on the unit physical density to obtain the function values of the structural flexibility, the total volume of the structural material and the structural overhang angle constraint function is:

[0043] Based on the physical density of the unit Finite element analysis is performed to obtain the structural flexibility c, and the unit physical density Sum up to get the total volume V of the structural material; calculate the structural overhang angle constraint function The function value of And it is used as the structural hanging angle constraint condition, and the structural hanging angle constraint condition is the structural hanging angle constraint function The function value of Less than or equal to the structural overhang angle constraint value ε:

[0044] The structural overhang angle constraint function The function value of The calculation is performed using the following formula:

[0045]

[0046] Among them A e is the unit area, which is a fixed constant under the same unit division; ρ i,j and are the density of unit (i, j) and the maximum support unit density among the three support units below it, i and j are the row and column numbers of the unit respectively; the parameter k controls the steepness of the function; the parameter α makes the function avoid the influence of density gradient area and approximation error by offset;

[0047] The maximum support unit density The following approximate function formula is used for calculation:

[0048]

[0049] Among them, the parameter p n Used to control the error of the approximate function.

[0050] In step 5), the method of performing sensitivity analysis on the function values of the structural flexibility, the total volume of the structural material, and the structural overhang angle constraint function, and filtering the sensitivity of the structural overhang angle constraint using the triangle directivity sensitivity filtering operator is:

[0051] The sensitivity analysis was performed using the chain rule, as follows:

[0052]

[0053] Where φ is the response variable to be analyzed, and the last two terms are the derivatives of the Heaviside density filtering and linear density filtering equations respectively; the structural flexibility c is the function of the unit physical density The sensitivity of the total volume V of the structural material to the unit physical density is calculated using the classical adjoint method formula. The sensitivity is 1;

[0054] For the structural overhang angle constraint function The function value of Sensitivity analysis of the structural overhang angle constraint of each unit is calculated as follows:

[0055]

[0056] The three terms on the right side of the equation are the unit physical density The influence of the change on the three supported units above can be calculated by the following formula:

[0057]

[0058] where s = j-1, j, j+1

[0059] Add two layers of empty cells on the left and right sides of the cell density matrix respectively. The above analysis can also be used for the boundary cells.

[0060] In order to promote the downward evolution of the structure and improve the stability of the optimization process, a triangle directivity sensitivity filtering operator is proposed. The sensitivity of the filtered structure overhang angle constraint is obtained by the weighted average of the sensitivities of all units in the triangular area above the unit e. The specific formula is as follows:

[0061]

[0062] H ei =max(0,R-Δ(e,i))

[0063] Among them, N e The unit set in the upper triangular area with a radius of R2 and the unit e as the center is filtered. ei is the weight coefficient between unit i and unit e, and Δ(e,i) is the distance between unit i and unit e.

[0064] In step 6), the optimization model is optimized and solved using an optimization algorithm. If the structural overhang angle constraint is not activated and the structural flexibility tends to be stable, the optimal printing direction is set to the direction where the function value of the structural overhang angle constraint function is minimized. The mesh is then redivided as required, and the method for activating the structural overhang angle constraint is as follows:

[0065] The optimization algorithm adopts a moving asymptote optimization algorithm; if the structural overhang angle constraint is not activated and the structural flexibility tends to be stable, the function value of the structural overhang angle constraint function under different printing directions is analyzed according to formula (1). Before the analysis, the design area is meshed according to the printing direction and the critical value of the structural overhang angle, and the cell density is obtained by linear interpolation according to the current design; after the analysis is completed, the optimal printing direction is set to the direction with the minimum function value of the structural overhang angle constraint function, and then the mesh is re-divided according to the demand, and the design variable value is updated;

[0066] The method for activating the structural overhang angle constraint is to set the structural overhang angle constraint value ε according to the following formula:

[0067]

[0068] Where loop0 and are the number of iterations when the constraint is activated and the function value of the structure overhang angle constraint function; loop is the number of loop iterations, and γ is a positive number greater than 1, which is used to control the decreasing rate of the structure overhang angle constraint value ε; as the number of iterations increases, the structure overhang angle constraint value ε decreases from gradually decreases and eventually approaches

[0069] The topology optimization method provided by the present invention, which takes into account different overhang angle constraints and printing directions in additive manufacturing, has the following beneficial effects:

[0070] Based on the framework of the variable density method (SIMP) method, an explicit and continuous structural overhang angle constraint condition was constructed. This constraint condition can be adaptively activated and tightened, and the optimization process is more stable. The design area is divided into a network using rectangular units, and the unit aspect ratio can be adjusted to be applicable to different structural overhang angle critical values (COA) situations. A triangle directivity sensitivity filtering operator is proposed to promote the downward evolution of the structure and improve stability. The established topology optimization method process is combined with the printing direction design, thereby significantly reducing the performance loss caused by the structural overhang angle constraint. Compared with the existing topology optimization design method considering the additive manufacturing structural overhang angle constraint, this method is applicable to different structural overhang angle constraint situations, and combined with the printing direction design, it can significantly reduce the performance loss caused by the structural overhang angle constraint. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 is a flow chart of a topology optimization method considering different overhang angle constraints and printing directions for additive manufacturing provided by the present invention;

[0072] Figure 2 It is a schematic diagram of grid division in the present invention;

[0073] Figure 3Schematic diagram of the sensitivity filtering of the triangle with the structure of the overhang angle constraint in the present invention, wherein Figure (a) is a schematic diagram of the sensitivity filtering area of the triangle, and Figure (b) is a schematic diagram of the sensitivity filtering effect;

[0074] Figure 4 is a structural design model in an embodiment of the present invention;

[0075] Figure 5 Schematic diagram of topology optimization results considering different overhang angle constraints and printing direction designs in an embodiment of the present invention. DETAILED DESCRIPTION

[0076] like Figure 1 As shown, the topology optimization design method considering the overhang angle constraint of the additive manufacturing structure provided by the present invention includes the following steps performed in sequence:

[0077] 1) Define the geometry of the design area, critical value of the overhang angle (COA), load and boundary conditions. According to the initial printing direction and critical value of the overhang angle, the design area is meshed using rectangular elements, such as Figure 2 As shown, based on the SIMP method framework, the design variables, optimization objectives and constraints are defined, and an optimization model is established. Thus, the unit density, material properties, structural material volume constraints, structural overhang angle constraint function, structural overhang angle constraint value, and optimization algorithm parameters are set, and the initial value of the structural overhang angle constraint value is made large to make it in an inactive state.

[0078] Define the geometry of the design area. Set the critical value of the overhang angle of the structure to the critical value of the overhang angle of the lower surface of the structure. Use rectangular cells to mesh the design area. Adjust the aspect ratio of the cells according to the initial printing direction and the critical value of the overhang angle of the structure so that the inclination angle of the cell diagonal relative to the substrate is the same as the critical value of the overhang angle of the structure, as shown in the following example: Figure 2 As shown; then, based on the SIMP method framework, the minimum structural flexibility is taken as the optimization goal, the discretized unit density and printing direction are used as design variables, and the function values of the structural material volume constraint and the structural overhang angle constraint function are used as constraints to establish an optimization model. The mathematical expression of the optimization model is:

[0079]

[0080] Where c is the structural flexibility;

[0081] d is the printing direction;

[0082] ρ is the cell density;

[0083] is the physical density of the unit after filtering;

[0084] U and K are the overall nodal displacement vector and overall stiffness matrix, respectively;

[0085] u e and k0 are the element displacement vector and element stiffness matrix respectively;

[0086] F is the external load vector of the structure;

[0087] N is the number of units;

[0088] V and V0 are the total volume of structural materials and the total volume of the design area, respectively;

[0089] V f is the volume constraint value of the structural material;

[0090] is the structural overhang angle constraint function;

[0091] ε is the structural overhang angle constraint value;

[0092] A e is the unit area;

[0093] is the density penalty function, Penalize the intermediate density, where p is the penalty factor, E min is the minimum elastic modulus, E0 is the unit elastic modulus;

[0094] In order to make the optimization process more stable, the initial value of the structural overhang angle constraint ε is made larger, so that the structural overhang angle constraint is always an invalid constraint in the early optimization process.

[0095] 2) Performing linear density filtering on the cell density in the above optimization model to obtain the cell intermediate density;

[0096] The unit intermediate density ρ * It is obtained by weighted average of the density of all cells within the circular range of the filter radius R. The specific formula is as follows:

[0097]

[0098] Among them, N e The unit set within the circular range with a filtering radius of R is centered on the unit e, H ei is the weight coefficient between unit i and unit e, and Δ(e,i) is the distance between unit i and unit e.

[0099] 3) Using the volume conservation type Heaviside function to filter the above unit intermediate density to obtain the unit physical density;

[0100] The volume conservation Heaviside function is used to calculate the cell intermediate density ρ* Filter to obtain unit physical density The specific formula is as follows:

[0101]

[0102] Among them, β is the control parameter; when the control parameter β is infinite, the cell intermediate density When the unit intermediate density is less than the parameter η 0; unit intermediate density When the unit intermediate density is greater than the parameter η is 1; the value of the parameter η is determined by bisection in each iteration to ensure that the volume of the structure remains unchanged before and after filtering.

[0103] 4) Based on the above-mentioned unit physical density, finite element analysis, volume constraint and structural overhang angle constraint response analysis are performed to obtain the function values of structural flexibility, total volume of structural materials and structural overhang angle constraint function;

[0104] Based on the physical density of the unit Finite element analysis is performed to obtain the structural flexibility c, and the unit physical density Sum up to get the total volume V of the structural material; calculate the structural overhang angle constraint function The function value of And it is used as the structural hanging angle constraint condition, and the structural hanging angle constraint condition is the structural hanging angle constraint function The function value of Less than or equal to the structural overhang angle constraint value ε:

[0105] The structural overhang angle constraint function The function value of The calculation is performed using the following formula:

[0106]

[0107] Among them A e is the unit area, which is a fixed constant under the same unit division; ρ i,j and are the density of unit (i, j) and the maximum support unit density among the three support units below it, i and j are the row and column numbers of the unit respectively; the parameter k controls the steepness of the function; the parameter α makes the function avoid the influence of density gradient area and approximation error by offset;

[0108] The maximum support unit density The following approximate function formula is used for calculation:

[0109]

[0110] Among them, the parameter p n Used to control the error of the approximate function.

[0111] 5) Perform sensitivity analysis on the function values of the above-mentioned structural flexibility, total volume of structural materials, and structural overhang angle constraint functions, and use a triangle directivity sensitivity filter operator to filter the sensitivity of the structural overhang angle constraint;

[0112] The sensitivity analysis was performed using the chain rule, as follows:

[0113]

[0114] Where φ is the response variable to be analyzed, and the last two terms are the derivatives of the Heaviside density filtering and linear density filtering equations respectively; the structural flexibility c is the function of the unit physical density The sensitivity of the total volume V of the structural material to the unit physical density is calculated using the classical adjoint method formula. The sensitivity is 1;

[0115] For the structural overhang angle constraint function The function value of Sensitivity analysis of the structural overhang angle constraint of each unit is calculated as follows:

[0116]

[0117] The three terms on the right side of the equation are the unit physical density The impact of changes on the three supported units above. Each item can be calculated using the following formula:

[0118]

[0119] where s = j-1, j, j+1

[0120] Add two layers of empty cells on the left and right sides of the cell density matrix respectively. The above analysis can also be used for the boundary cells.

[0121] like Figure 3 As shown in (a), in order to promote the downward evolution of the structure and improve the stability of the optimization process, a triangle directivity sensitivity filtering operator is proposed. The sensitivity of the filtered structure's overhang angle constraint is obtained by the weighted average of all unit sensitivities in the triangular area above unit e. The specific formula is as follows:

[0122]

[0123] H ei =max(0,R-Δ(e,i))

[0124] Among them, N eThe unit set in the upper triangular area with a radius of R2 and the unit e as the center is filtered. ei is the weight coefficient between unit i and unit e, and Δ(e,i) is the distance between unit i and unit e.

[0125] Figure 3 (b) is a schematic diagram of the sensitivity filtering effect. It can be seen that the overhang angle constraint sensitivity of all units in the triangular area below unit e is affected by unit e.

[0126] 6) Use the optimization algorithm to optimize and solve the optimization model. If the structural overhang angle constraint is not activated and the structural flexibility tends to be stable, set the optimal printing direction to the direction with the minimum function value of the structural overhang angle constraint function, then re-divide the mesh as needed and activate the structural overhang angle constraint; otherwise, go directly to step 7);

[0127] The optimization algorithm adopts the moving asymptote optimization algorithm (MMA); if the structural overhang angle constraint is not activated and the structural flexibility tends to be stable, the function value of the structural overhang angle constraint function under different printing directions is analyzed according to formula (1). Before the analysis, the design area is meshed according to the printing direction and the critical value of the structural overhang angle, and the cell density is obtained by linear interpolation according to the current design; after the analysis is completed, the optimal printing direction is set to the direction with the minimum function value of the structural overhang angle constraint function, and then the mesh is re-divided according to the demand, and the design variable value is updated;

[0128] The method for activating the structural overhang angle constraint is to set the structural overhang angle constraint value ε according to the following formula:

[0129]

[0130] Where loop0 and are the number of iterations when the constraint is activated and the function value of the structure overhang angle constraint function; loop is the number of loop iterations, and γ is a positive number greater than 1, which is used to control the decreasing rate of the structure overhang angle constraint value ε; as the number of iterations increases, the structure overhang angle constraint value ε decreases from gradually decreases and eventually approaches

[0131] 7) Update the unit density and the structural overhang angle constraint value, and determine whether it has converged. If not, return to step 2) to iterate the algorithm. If it has converged, the optimization ends, and the topology optimization result is finally output, and the printing direction is given.

[0132] The following is a further explanation of the topological optimization design of the MBB beam structure. The design domain is a 150×50 rectangle. The horizontal displacement of the left side and the vertical displacement of the lower right corner are constrained, and the downward concentrated force acts on the upper left corner, as shown in the following figure. Figure 4As shown in Figure 2. L, B, R, and U are four potential printing directions. The parameters are set as follows: unit elastic modulus E0 = 1, minimum elastic modulus E min = 1e-9, Poisson's ratio v = 0.3. The load size is unit load 1. The overhang angle constraint function parameters are k = 150 and α = 0.03. The SIMP interpolation parameter p is 3, p n = 60. γ and δ were set to 1.3 and 0.1, respectively. The optimization started with a uniform distribution density of 0.5, and the motion limits of the design variables were all 0.1. The Heaviside parameter was initially set to 1 and doubled every 50 iterations. The cell width was fixed to unit length, while the cell height was varied to accommodate different COAs. Topology optimization was performed for COAs of 30°, 45°, and 60°, and for filter radii R of 3.5 and 4.5.

[0133] Figure 5 (a)-(f) show the topology optimization design results under different COA and filter radius R, where the outer square represents the optimal printing position of the substrate. The corresponding structural flexibility c and the structural flexibility c without considering the structural overhang angle constraint are also given. ref ratio.

[0134] It will be easily understood by those skilled in the art that the specific implementation cases described above are merely examples for the present invention and are not intended to limit the present invention. Any modifications, replacements and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A topology optimization method considering different overhang angle constraints and printing directions for additive manufacturing, characterized by: The topology optimization method comprises the following steps performed in sequence: 1) Define the geometry of the design area, critical value of the structural overhang angle, load, and boundary conditions. Based on the initial printing direction and critical value of the structural overhang angle, mesh the design area using rectangular elements. Then, based on the SIMP framework, define the design variables, optimization objectives, and constraints, and establish an optimization model. The element density, material properties, structural material volume constraints, structural overhang angle constraint function, structural overhang angle constraint value, and optimization algorithm parameters are set. The initial value of the structural overhang angle constraint is set large to keep it in an inactive state. 2) Performing linear density filtering on the cell density in the above optimization model to obtain the cell intermediate density; 3) Using the volume conservation type Heaviside function to filter the above unit intermediate density to obtain the unit physical density; 4) Based on the above-mentioned unit physical density, finite element analysis, volume constraint and structural overhang angle constraint response analysis are performed to obtain the function values of structural flexibility, total volume of structural materials and structural overhang angle constraint function; 5) Perform sensitivity analysis on the function values of the above-mentioned structural flexibility, total volume of structural materials, and structural overhang angle constraint functions, and use a triangle directivity sensitivity filter operator to filter the sensitivity of the structural overhang angle constraint; 6) Use the optimization algorithm to optimize and solve the optimization model. If the structural overhang angle constraint is not activated and the structural flexibility tends to be stable, set the optimal printing direction to the direction with the minimum function value of the structural overhang angle constraint function, then re-divide the mesh as needed and activate the structural overhang angle constraint; otherwise, go directly to step 7); 7) Update the unit density and the structural overhang angle constraint value, and determine whether it has converged. If not, return to step 2) to iterate the algorithm. If it has converged, the optimization ends, and the topology optimization result is finally output, and the printing direction is given; In step 4), the method for performing finite element analysis, volume constraint and structural overhang angle constraint response analysis based on the unit physical density to obtain the function values of the structural flexibility, the total volume of the structural material and the structural overhang angle constraint function is: Based on the physical density of the unit Finite element analysis is performed to obtain the structural flexibility c, and the unit physical density Sum up to get the total volume V of the structural material; calculate the structural overhang angle constraint function The function value of And it is used as the structural hanging angle constraint condition, and the structural hanging angle constraint condition is the structural hanging angle constraint function The function value of Less than or equal to the structural overhang angle constraint value ε: The structural overhang angle constraint function The function value of The calculation is performed using the following formula: Among them A e is the unit area, which is a fixed constant under the same unit division; ρ i,j and are the density of unit (i, j) and the maximum support unit density among the three support units below it, i and j are the row and column numbers of the unit respectively; the parameter k controls the steepness of the function; the parameter α makes the function avoid the influence of density gradient area and approximation error by offset; The maximum support unit density The following approximate function formula is used for calculation: Among them, the parameter p n Used to control the error of the approximate function.

2. The topology optimization method considering different overhang angle constraints and printing directions for additive manufacturing according to claim 1, characterized in that: In step 1), the geometric shape of the design area, the critical value of the structural overhang angle (COA), the load and the boundary conditions are defined, the design area is meshed using rectangular elements according to the initial printing direction and the critical value of the structural overhang angle, and then the design variables, optimization objectives and constraints are defined based on the SIMP method framework to establish an optimization model. The element density, material properties, structural material volume constraints, structural overhang angle constraint function, structural overhang angle constraint value, and optimization algorithm parameters are set, and the method of making the initial value of the structural overhang angle constraint value large so that it is in an inactive state is as follows: The geometric shape of the design area is defined, and the critical value of the structure's overhang angle is set to the critical value of the overhang angle of the lower surface of the structure. The design area is meshed using rectangular elements, and the aspect ratio of the elements is adjusted according to the initial printing direction and the critical value of the structure's overhang angle, so that the inclination angle of the element diagonal relative to the substrate is the same as the critical value of the structure's overhang angle. Then, based on the SIMP method framework, with minimizing structural flexibility as the optimization goal, the discretized element density and printing direction are used as design variables, and the function values of the structural material volume constraint and the structural overhang angle constraint function are used as constraints to establish an optimization model. The mathematical expression of the optimization model is: Where c is the structural flexibility; d is the printing direction; ρ is the cell density; is the physical density of the unit after filtering; U and K are the overall nodal displacement vector and overall stiffness matrix, respectively; u e and k0 are the element displacement vector and element stiffness matrix respectively; F is the external load vector of the structure; N is the number of units; V and V0 are the total volume of structural materials and the total volume of the design area, respectively; V f is the volume constraint value of the structural material; is the structural overhang angle constraint function; ε is the structural overhang angle constraint value; A e is the unit area; is the density penalty function, Penalize the intermediate density, where p is the penalty factor, E min is the minimum elastic modulus, E0 is the unit elastic modulus; In order to make the optimization process more stable, the initial value of the structural overhang angle constraint ε is made larger, so that the structural overhang angle constraint is always an invalid constraint in the early optimization process.

3. The topology optimization method considering different overhang angle constraints and printing directions for additive manufacturing according to claim 1, characterized in that: In step 2), the method of performing linear density filtering on the unit density in the optimized model to obtain the unit intermediate density is: The unit intermediate density ρ * It is obtained by weighted average of the density of all cells within the circular range of the filter radius R. The specific formula is as follows: Among them, N e The unit set within the circular range with a filtering radius of R is centered on the unit e, H ei is the weight coefficient between unit i and unit e, and Δ(e,i) is the distance between unit i and unit e.

4. The topology optimization method considering different overhang angle constraints and printing directions for additive manufacturing according to claim 1, characterized in that: In step 3), the method of filtering the unit intermediate density using the volume conservation type Heaviside function to obtain the unit physical density is: The volume conservation Heaviside function is used to calculate the cell intermediate density ρ * Filter to obtain unit physical density The specific formula is as follows: Among them, β is the control parameter; when the control parameter β is infinite, the cell intermediate density When the unit intermediate density is less than the parameter η 0; unit intermediate density When the unit intermediate density is greater than the parameter η is 1; the value of the parameter η is determined by bisection in each iteration to ensure that the volume of the structure remains unchanged before and after filtering.

5. The topology optimization method considering different overhang angle constraints and printing directions in additive manufacturing according to claim 1, characterized in that: In step 5), the method of performing sensitivity analysis on the function values of the structural flexibility, the total volume of the structural material, and the structural overhang angle constraint function, and filtering the sensitivity of the structural overhang angle constraint using the triangle directivity sensitivity filtering operator is: The sensitivity analysis was performed using the chain rule, as follows: Where φ is the response variable to be analyzed, and the last two terms are the derivatives of the Heaviside density filtering and linear density filtering equations respectively; the structural flexibility c is the function of the unit physical density The sensitivity of the total volume V of the structural material to the unit physical density is calculated using the classical adjoint method formula. The sensitivity is 1; For the structural overhang angle constraint function The function value of Sensitivity analysis of the structural overhang angle constraint of each unit is calculated as follows: The three terms on the right side of the equation are the unit physical density The influence of the change on the three supported units above can be calculated by the following formula: where s = j-1, j, j+1 Add two layers of empty cells on the left and right sides of the cell density matrix respectively. The above analysis can also be used for the boundary cells. In order to promote the downward evolution of the structure and improve the stability of the optimization process, a triangle directivity sensitivity filtering operator is proposed. The sensitivity of the filtered structure overhang angle constraint is obtained by the weighted average of the sensitivities of all units in the triangular area above the unit e. The specific formula is as follows: Among them, N e The unit set in the upper triangular area with a radius of R2 and the unit e as the center is filtered. ei is the weight coefficient between unit i and unit e, and Δ(e,i) is the distance between unit i and unit e.

6. The topology optimization method considering different overhang angle constraints and printing directions in additive manufacturing according to claim 1 or 2, characterized in that: In step 6), the optimization model is optimized and solved using an optimization algorithm. If the structural overhang angle constraint is not activated and the structural flexibility tends to be stable, the optimal printing direction is set to the direction where the function value of the structural overhang angle constraint function is minimized. The mesh is then redivided as required, and the method for activating the structural overhang angle constraint is as follows: The optimization algorithm adopts a moving asymptote optimization algorithm; if the structural overhang angle constraint is not activated and the structural flexibility tends to be stable, the function value of the structural overhang angle constraint function under different printing directions is analyzed according to formula (1). Before the analysis, the design area is meshed according to the printing direction and the critical value of the structural overhang angle, and the cell density is obtained by linear interpolation according to the current design; after the analysis is completed, the optimal printing direction is set to the direction with the minimum function value of the structural overhang angle constraint function, and then the mesh is re-divided according to the demand, and the design variable value is updated; The method for activating the structural overhang angle constraint is to set the structural overhang angle constraint value ε according to the following formula: Where loop0 and are the number of iterations when the constraint is activated and the function value of the structure overhang angle constraint function; loop is the number of loop iterations, and γ is a positive number greater than 1, which is used to control the decreasing rate of the structure overhang angle constraint value ε; as the number of iterations increases, the structure overhang angle constraint value ε decreases from gradually decreases and eventually approaches

Images