Multi-axis 3D printing two-step optimization design and manufacturing method

Abstract

This invention relates to a two-step optimization design and manufacturing method for multi-axis 3D printing, comprising the following steps: obtaining the optimal structure through topology optimization without overhang constraints; then dividing the printing domain into printing regions and calculating the locally optimal printing direction for each region; using the locally optimal printing direction as the unit overhang angle constraint and adding a unit horizontal neighborhood unit density constraint term to avoid overhang characteristics in the optimization process; performing integrated topology optimization with angle constraints, while simultaneously conducting sensitivity analysis, and manufacturing a self-supporting multi-axis 3D printed structure. The beneficial effects of this invention are: it adopts a two-step optimization design progressive mode; it solves the problems of printing region division, locally optimal printing direction, and invalid boundary unit handling in the design domain; it considers overhang constraints, using the locally optimal printing direction as the printing direction for each unit in the region, suppressing unprintable units, and thus obtaining a self-supporting structure.

Description

Technical Field

[0001] This invention belongs to the fields of structural engineering and additive manufacturing technology, and particularly relates to a two-step optimized design and manufacturing method for multi-axis 3D printing. Multi-axis refers to a 3D printing system with more than three rotating axes, including the printer head rotating axis and the positioning device rotating axis. Background Technology

[0002] Compared to traditional manufacturing processes, 3D printing technology offers advantages such as high efficiency and high precision, making it more suitable for processing and manufacturing complex structures. As modern engineering structures become increasingly complex and personalized, the demand for 3D printing of complex structural components continues to rise. Traditional structural design methods often fall short in achieving this, and topology optimization provides an effective solution. Structures optimized for topology exhibit excellent mechanical properties and a reasonable material distribution, but their geometry is often more complex.

[0003] 3D printing still requires structures to meet specific manufacturing constraints to ensure successful printing. Among these constraints, the sag effect caused by gravity is a major one. The sag effect occurs when the angle between the structure's boundary and the horizontal plane is less than a critical value; due to gravity, the material deposition process can collapse, affecting the printing quality and even causing printing failure. An effective way to overcome the sag effect is to introduce angular constraints in the structural optimization design to obtain a structure with optimal mechanical properties that meets these angular manufacturing constraints. Current research mainly focuses on 3-axis 3D printing using robotic arms. While this can produce self-supporting structures that meet manufacturing constraints, avoiding the need for supports, it often leads to a significant increase in material usage and a substantial decrease in structural performance.

[0004] Multi-axis 3D printing technology is widely used in aerospace, automotive engineering, and other fields. Because multi-axis 3D printers have a freely rotating base, they can dynamically adjust the printing direction during the printing process to avoid overhang effects, effectively solving the problems of increased volume and significant performance degradation in self-supporting structures for 3-axis 3D printing. Therefore, the rational and effective integration of multi-axis 3D printing manufacturing with the optimized design of self-supporting structures is a crucial factor in the integrated optimized design and manufacturing of complex structures.

[0005] In the design of self-supporting structures based on multi-axis 3D printing, three main problems need to be solved: 1) how to divide the design domain into different printing areas; 2) how to obtain the optimal printing direction for each partition; and 3) how to handle unprintable units after changing the printing direction of each partition.

[0006] In conclusion, it is essential to study an optimized design and manufacturing method for multi-axis 3D printing to achieve the optimal structural configuration design and integrated manufacturing of multi-axis 3D printing at any tilt angle. Summary of the Invention

[0007] The purpose of this invention is to overcome the shortcomings of the prior art and provide a two-step optimized design and manufacturing method for multi-axis 3D printing.

[0008] This two-step optimization design and manufacturing method for multi-axis 3D printing includes the following steps:

[0009] S1, First step design: First, use topology optimization without overhang constraints to obtain the optimal structure, then divide the printing area, and calculate the local optimal printing direction for different printing areas; if the unprintable ratio is less than the threshold, terminate the optimization and proceed to step S3, otherwise proceed to step S2.

[0010] S2. Second step design: Considering the overhang constraint, the local optimal printing direction is used as the unit overhang angle constraint, and the unit horizontal neighborhood unit density constraint term is added to avoid the overhang characteristics of the optimization process; integrated topology optimization with angle constraints is used to perform integrated topology optimization, and sensitivity analysis is performed to suppress unprintable units, thereby obtaining a self-supporting structure;

[0011] S3, Multi-axis 3D Printing Manufacturing: Extract and optimize structural information, establish a 3D solid model, divide and slice the structure and generate a printing path to carry out multi-axis 3D printing manufacturing of self-supporting structures.

[0012] Preferably, in step S1, the method for topology optimization without overhang constraints specifically involves using a density-based SIMP model, considering free form, and setting the density ρ = ρ1, ρ2, ..., ρ of each element within the design domain involved in topology optimization. nele As a design variable, the expression for structural topology optimization is:

[0013]

[0014] In the formula, U is the global displacement vector; F is the global nodal load vector; K is the total stiffness matrix; the objective function C(ρ) is the total strain energy under external force; v i Let f be the volume of the i-th unit; f is the spatial proportion; the density ρ of each unit in the design domain involving topology optimization takes values ​​from 0 to 1.

[0015] As a preferred option, in step S1: the printing sub-region is divided by mesh discretizing the entire design domain or by using a manual division method; the manual division method specifically involves using Harris or SUSAN corner detection algorithms to obtain the structural corners of simple structures, and using the corners as vertices of the sub-regions to divide them, and the resulting rectangular sub-regions are the printing sub-regions.

[0016] Preferably, in step S1: the optimal printing direction for each printing sub-region is determined by the tilt direction of the boundary cells in the printing sub-region. A boundary cell is a cell with empty cells in its adjacent set. The expression for determining a boundary cell is...

[0017]

[0018] In the formula, The value is used to characterize whether an element is a boundary element. When an element is a boundary element, its... The value is 1, when a certain cell is an internal cell. The value is 0;

[0019] The density gradient direction of the cells is obtained by using convolution kernels, the tilt direction of the boundary cells is obtained by transformation, and the printability of the boundary cells in each printing sub-region is considered to determine the local optimal printing direction of each printing sub-region.

[0020] The cell density gradient direction is obtained using a convolution kernel. The cell density gradient direction is described as follows:

[0021]

[0022] In the formula, x j y i The coordinate position of the element is defined; edge-additional elements are introduced into the design domain edge, with the density of edge-additional elements adjacent to the base set to 1, and the density of edge-additional elements in other design domains set to 0;

[0023] The element density gradient direction is converted into the tilt direction of the structural boundary, which is the angle between the structural boundary and the x-axis, and is orthogonal to the element density gradient direction. For element density gradient directions in different quadrants, the corresponding structural boundary tilt directions are:

[0024]

[0025] In the formula, The direction of inclination of the structural boundary between 0 and π. This is the unit density gradient vector.

[0026] As a preferred approach: all elements adjacent to the base are designated as invalid boundary elements, which also constitute the first layer of the design domain; when non-printable elements appear in the i-th layer in the vertical printing direction, all elements below the i-th layer are considered invalid boundary elements; elements satisfying the following formula are also considered invalid boundary elements.

[0027] b i ≤ω|a i | (5)

[0028] In the formula, The allowable rotation angle of the base is generally taken as... θ is the maximum overhang angle; x j y j This refers to the unit coordinate position; The value is used to characterize whether an element is a boundary element. When an element is a boundary element, its... The value is 1, when a certain cell is an internal cell. The value is 0;

[0029] The printing direction of invalid boundary elements was adjusted to the vertical direction after linear optimization.

[0030] Preferably, in step two: the degree to which the element violates the overhang angle constraint is calculated, and the expression for the degree to which the element violates the overhang angle constraint is:

[0031]

[0032] In the formula, t i The value represents the degree of violation of the overhang angle constraint, and cosθ is the cosine value of the critical overhang angle of the structure. The unit boundary normal vector; when t i If the tolerance is less than or equal to δ, then the unit satisfies the suspension angle constraint, and the range of δ is 0 < δ ≤ 0.001;

[0033] The tilt direction of the structural boundary of each sub-region is obtained from equation (4). Considering the overhang angle of the boundary unit and avoiding printing collisions between adjacent areas as constraints, the local optimal printing direction of each printing sub-region is calculated.

[0034]

[0035] In the formula, The local optimal printing direction vector for printing sub-regions; O represents the degree vector of structural elements violating the overhang angle constraint; V is 1 for effective boundary elements and 0 for other boundary elements; M is a 0-1 mapping matrix; is the tilt direction vector of the structural unit; and Defined as the printing direction of two adjacent sub-regions. The maximum allowable deflection angle; when the tilt direction vector of the structural element... Within the printable range, the value of O is... Conversely, the value of O is greater than

[0036] The objective function is rewritten as:

[0037] V io =OT V+PQ (8)

[0038] In the formula, P is the weight of the penalty function, which is taken as 0.01; the expression for Q is:

[0039]

[0040] In the formula, It is the angle between the vertical direction and the x-axis.

[0041] As a preferred option, in step two: the expression for structural topology optimization under the linear angle constraints of the units within each printing partition is as follows:

[0042]

[0043] In the formula, γ is a constraint term for the density of cells in the horizontal neighborhood of a given cell. i These are the parameter values ​​for the overhang characteristics of the structural boundary elements after topology optimization; This is a linear angle constraint term for the locally optimal printing direction in a given region. The parameters characterizing the locally optimal printing direction of an element are determined by the element's overhang angle; U is the global displacement vector; F is the global nodal load vector; K is the total stiffness matrix; the objective function C(ρ) is the total strain energy under external force; v i Let f be the volume of the i-th unit; f be the spatial proportion; and ρ be the density of each unit, with a value ranging from 0 to 1.

[0044] As a preferred option, in step two: the sensitivity of the objective function to the design variables is calculated according to the chain rule in order to solve the stability problem of the optimal solution;

[0045] For the objective function:

[0046]

[0047] For the constraint of the hanging angle:

[0048]

[0049] For suspension characteristic constraints:

[0050]

[0051] In the formula, h(x) is the Sigmoid function, and t il and t ir τ represents the degree to which the left and right boundaries violate the overhang angle constraint, respectively. il and τ ir These represent the degree to which the left and right boundaries violate the suspended characteristic constraints, respectively; u kLet k be the displacement vector of the k-th element, k0 be the initial element stiffness matrix, E0 be the Young's modulus of the material, and E min The parameter is close to 0, p is the penalty factor, and is generally taken as p = 3; v i Let ρ be the volume of the i-th unit; ρ is the density of each unit, with a value ranging from 0 to 1.

[0052] As a preferred option, step S3 specifically involves: performing 3D modeling using Rhino software; slicing the solid model obtained from the 3D modeling using Cura software and generating printing paths for multi-axis 3D printing manufacturing.

[0053] The beneficial effects of this invention are:

[0054] 1) This invention provides a two-step optimization design and manufacturing method for multi-axis 3D printing, which adopts a two-step optimization design progressive mode. The first step of the design solves the problems of printing sub-region division, local optimal printing direction and invalid boundary element handling in the design domain. The second step of the design considers the overhang constraint, takes the local optimal printing direction as the printing direction of each element in the sub-region, and uses integrated topology optimization to suppress unprintable elements to obtain a self-supporting structure.

[0055] 2) The multi-axis 3D printing two-step optimization design and manufacturing method provided by this invention combines two-step optimization design and multi-axis 3D printing manufacturing. Based on the first step design, the optimal configuration without overhang constraints is obtained. By using corner detection, unit tilt angle, and preprocessing, the printing area is divided, the local optimal printing direction is achieved, and invalid boundary units are printed. Based on the second step design, the integrated optimization design with overhang angle constraints is obtained. Through multi-axis partitioned 3D printing of the print head and base rotation axis, the optimal configuration of the self-supporting structure under any tilt angle is achieved. Through 3D solid model partitioning and slicing and printing path generation, the integrated design and manufacturing of the optimal configuration of the complex self-supporting structure is achieved.

[0056] 3) This invention is based on multi-axis partitioned 3D printing that takes into account the rotation axis of the print head and the base. During the printing process, the printing direction is dynamically adjusted to avoid the overhang effect of the structure during printing, so as to achieve the optimal configuration printing of self-supporting structure under any tilt angle. Through angle constraint integrated optimization, the overall optimization design and printing of insufficiently printed areas are realized, which effectively solves the problems of increased volume and significant performance degradation when printing self-supporting structures in 3D. Attached Figure Description

[0057] Figure 1 This is a flowchart illustrating the two-step optimization design and manufacturing method for multi-axis 3D printing of the present invention.

[0058] Figure 2a This is a schematic diagram of the rectangular design domain mesh discretization;

[0059] Figure 2bThis is a schematic diagram of a rectangular design domain divided into discrete mesh regions.

[0060] Figure 3a This is a schematic diagram of structural corner detection;

[0061] Figure 3b This is a schematic diagram of the design domain divided into artificial areas;

[0062] Figure 4a This is a schematic diagram of the cell neighborhood pattern;

[0063] Figure 4b This is a schematic diagram of the boundary extension design domain of the additional unit;

[0064] Figure 5 This is a schematic diagram showing the orientation of buildings in different areas;

[0065] Figure 6a This is a schematic diagram of the neighborhood pattern of the boundary unit;

[0066] Figure 6b A schematic diagram of the neighborhood pattern of non-boundary units;

[0067] Figure 7 This is a diagram illustrating a collision that occurred during the printing process;

[0068] Figure 8 This is a schematic diagram of the local optimal printing direction for adjacent sub-regions;

[0069] Figure 9 This is a schematic diagram of the invalid boundary elements adjacent to the base.

[0070] Figure 10 This is a schematic diagram of the process of scanning layer by layer to the non-printable cell;

[0071] Figure 11a This is a schematic diagram of the printable unit when the base rotates to 0.

[0072]

[0073]

[0074] Figure 12 This is a schematic diagram showing the printable range of different units;

[0075] Figure 13a This is a schematic diagram of the design domain for a cantilever beam;

[0076] Figure 13b This is a schematic diagram of the optimal topology configuration of the cantilever beam;

[0077] Figure 13cThis is a schematic diagram of the design domain division for a cantilever beam.

[0078] Figure 13d This is a printed schematic diagram of the cantilever beam. Detailed Implementation

[0079] The present invention will be further described below with reference to embodiments. The description of the embodiments below is only for the purpose of helping to understand the present invention. It should be noted that those skilled in the art can make several modifications to the present invention without departing from the principle of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

[0080] Example 1

[0081] As one example, such as Figure 1 As shown, the two-step optimization design and manufacturing method for multi-axis 3D printing includes the following steps:

[0082] S1. First step design: Optimal structure is obtained through topology optimization without overhang constraints, leading to a printing scheme, including printing region division and locally optimal printing direction; if the non-printable ratio (i.e., the ratio of non-printable boundary cells to all boundary cells) is less than the threshold σ... r If the optimization is successful, the multi-axis 3D printing manufacturing will be terminated directly; otherwise, the design process will proceed to step S2, the second step of the design process.

[0083] S1.1 Topology optimization without overhang constraints: A density-based SIMP model is used as the topology optimization method, considering free form, and the density of each element in the design domain is ρ = ρ1, ρ2, ..., ρ nele As a design variable, the expression for structural topology optimization is:

[0084]

[0085] In the formula, the first equation represents the design variable, the second equation represents the objective function, and the third to fifth equations represent the constraints; U is the global displacement vector; F is the global nodal load vector; K is the overall stiffness matrix; the objective function C(ρ) is the total strain energy under external force; v i Let f be the volume of the i-th unit; f is the spatial proportion; the unit density ρ takes a value between 0 and 1.

[0086] S1.2, Printing divided into regions: such as Figures 2a-2b As shown, one way to divide the design domain into regions is to discretize the entire design domain using a mesh. The mesh density must be limited to an appropriate range because if there are too many boundary cells in a region, it will increase the difficulty of finding common angle ranges; conversely, if the region size is too small, more empty regions will appear.

[0087] like Figures 3a-3b As shown, another way to divide the design domain is by manual division; corner detection algorithms such as Harris and SUSAN are used to obtain the structural corners of simple structures, and the corners are used as vertices of rectangular regions to obtain reliable division results;

[0088] S1.3, Local Optimal Printing Direction: The density gradient direction of the cell is obtained by using a convolution kernel, and the tilt direction of the boundary cell is obtained by transformation. The printability of the boundary cell is considered to determine the local optimal printing direction of the sub-region.

[0089] like Figure 4a As shown, the cell density gradient direction is obtained using a convolution kernel (neighborhood); the cell density gradient direction is described as follows:

[0090]

[0091] In the formula, x j y j This represents the unit coordinate position.

[0092] like Figure 4b As shown, in order to consider the external boundary elements, additional edge elements need to be introduced in the design domain edge; the density of the additional edge elements adjacent to the base is set to 1, and the density of other additional edge elements is set to 0.

[0093] To avoid nonlinear issues during the process of obtaining the locally optimal printing direction, the element density gradient direction needs to be further converted into the tilt direction of the structural boundary. This tilt direction is defined as the angle between the structural boundary and the x-axis. The tilt direction is orthogonal to the element density gradient direction. For the element density gradient direction in different quadrants, the element tilt angle is:

[0094]

[0095] In the formula, The tilt angle of the unit between 0 and π. The unit density gradient vector;

[0096] like Figure 5 As shown, after determining the printing sub-regions in step S1.2, the optimal printing direction for each sub-region is determined by the tilt direction of the boundary cells in the sub-region; as long as all boundary cells can be printed, all cells in each region can be printed; the building direction of the sub-region, i.e., the locally optimal printing direction, can be determined by considering the printability of the boundary cells.

[0097] like Figures 6a-6b As shown, to make the boundary outline clear, cells with empty cells in adjacent sets are determined to be boundary cells. The determination expression is:

[0098]

[0099] In the formula, The value represents the boundary element; the value is 1 for the boundary element and 0 for the inner element.

[0100] like Figure 7 As shown, when the substrate changes its tilt angle during printing, if the curvature of each layer exceeds a specific value, the print head and the structure will collide; the printing direction of two adjacent areas should be greater than the maximum allowable deflection angle. To avoid collisions, the expression is:

[0101]

[0102] In the formula, Defined as the printing direction of two adjacent sub-regions, such as Figure 8 As shown;

[0103] like Figure 9 As shown, elements that do not affect the local optimal printing direction of the sub-region are defined as invalid boundary elements and require preprocessing. For example, elements adjacent to the base can be completely printed along the vertical printing direction, which are important components of this type of invalid boundary elements and do not need to be considered in multi-axis optimization.

[0104] The expression for an element that satisfies the suspension angle constraint is:

[0105]

[0106] In the formula, δ represents the tolerance, with a value range of 0 < δ ≤ 0.001, and t i The value represents the degree to which the overhang angle constraint is violated;

[0107] like Figure 10 As shown, the scanning process starts from the first layer of the design domain and proceeds along the vertical printing direction until an unprintable cell appears in the i-th layer; at this point, cells below the i-th layer are considered invalid boundary cells.

[0108] like Figures 11a-11c As shown, the elements that satisfy equation (7) are also considered as invalid boundary elements, and these elements do not need to be considered in multi-axis optimization; because their element density gradient direction is always within the allowable range of the base rotation range -π / 2 to π / 2.

[0109] b i ≤ω|a i | (7)

[0110] In the formula, The allowable rotation angle of the base is generally taken as... θ is the maximum overhang angle; a j bj The expression for this is detailed in equation (2);

[0111] Invalid boundary elements have no impact on the region division method. After linear optimization, the printing direction of invalid boundary elements can be adjusted to the vertical direction to simplify the optimization. Considering the collision problem (5), the overhang angle of the boundary elements obtained from equations (2) to (3) is substituted into equation (8) as a constraint term to obtain the local optimal printing direction of each region:

[0112]

[0113] In the formula, is the local optimal printing direction vector for the sub-region; O represents the degree vector of the structural unit violating the overhang angle constraint; V is 1 for effective boundary units and 0 for other boundary units; M is a 0-1 mapping matrix; is the tilt direction vector of the structural unit; This is the maximum deflection angle;

[0114] like Figure 12 As shown, in equation (8), if the tilt direction vector of the structural unit Within the printable range, the value of O is... Conversely, the value of O is greater than This is a linear programming problem, which can be solved using linear solvers such as Mosek.

[0115] To reduce the likelihood of collisions, the printing direction should be tilted as much as possible towards the vertical direction. The objective function can be rewritten as follows:

[0116] V io =O T V+PQ (9)

[0117] In the formula, P is the weight of the penalty function, which is taken as 0.01; the expression for Q is:

[0118]

[0119] In the formula, The angle between the vertical direction and the x-axis;

[0120] S2. Second step design: Considering the overhang constraint, the local optimal printing direction is used as the printing direction of each unit in the sub-region. Integrated topology optimization is used to suppress unprintable units to obtain a self-supporting structure.

[0121] S2.1 Adding overhang angle constraints: The local optimal printing direction of the sub-region is used as the overhang angle constraint of the unit, and the unit horizontal neighborhood unit density constraint term is added to avoid the overhang characteristics of the optimization process;

[0122] After obtaining the locally optimal printing direction for each region, if there are still units that violate the overhang feature constraint, it is necessary to optimize the structure as a whole.

[0123] The tilt direction of the boundary cells in each partition determines the optimal local printing direction for each partition, and the constraint terms. The parameter values ​​used in the text represent the overhang angle of the unit by employing the local printing direction of the structure. Constraints The parameter γ, representing the overhang characteristics of the boundary elements of the topology-optimized structure, is used to characterize the density of the horizontal neighborhood elements of the element. i .

[0124] S2.2 Integrated Topology Optimization: Perform integrated topology optimization with angle constraints and conduct sensitivity analysis;

[0125] The expression considering the linear angle constraints of cells within each print partition is as follows:

[0126]

[0127] In the formula, This is a constraint term for the density of cells in the horizontal neighborhood of a given cell. This is a linear angle constraint term for the locally optimal printing direction in a given region. The parameters are used to characterize the local optimal printing direction of the unit (in the case of overhang angle). Other parameters are the same as in equation (1).

[0128] Sensitivity analysis is often accompanied by topology optimization. Adding holes or removing material in the structure is actually considering the sensitivity of the objective function in that region. In topology optimization, the sensitivity of the objective function to the design variables, i.e., the partial derivatives, is solved according to the chain rule to solve the stability problem of the optimal solution.

[0129] For the objective function:

[0130]

[0131] For the constraint of the hanging angle:

[0132]

[0133] For suspension characteristic constraints:

[0134]

[0135] In the formula, h(x) is the Sigmoid function, and t il t ir τ represents the magnitude of the violation of the critical overhang angle at the left and right boundaries, respectively. il τ irThese represent the magnitudes of the suspension feature violation at the left and right boundaries, respectively; uk is the displacement vector of the k-th element, k0 is the initial element stiffness matrix, E0 is the Young's modulus of the material, and E... min The parameter is close to 0, and p is the penalty factor, which is usually taken as p = 3.

[0136] S3, Multi-axis 3D Printing Manufacturing: Optimized structural information is extracted from the optimization results. After component assembly and node generation, a 3D solid model is built using Rhino software. The solid model is then partitioned and sliced ​​using Cura software to generate printing paths, enabling the multi-axis 3D printing manufacturing of self-supporting structures.

[0137] Example 2

[0138] As another embodiment, based on the two-step optimization design and manufacturing method for multi-axis 3D printing proposed in Embodiment 1, this embodiment proposes a test embodiment for the two-step optimization design and manufacturing of a cantilever beam model for multi-axis 3D printing to verify the effectiveness of the method of the present invention.

[0139] like Figure 13a As shown, the design domain uses a cantilever beam model with a length of 120 and a height of 60. The material's elastic modulus is 1.0, the volume fraction constraint is 0.3, and the penalty coefficient is 3. The left end is fixed, and the load is applied at the midpoint of the beam's right boundary. After topology optimization, the optimal topology is obtained as shown. Figure 13b As shown, the final objective function value after optimization is 121.03; the cantilever beam printing region division and cantilever beam printing curve are shown below. Figure 13c , Figure 13d As shown.

[0140] In this embodiment, after considering multi-axis topology optimization, all structural units can be printed completely without the need for integrated topology optimization. This solves the problem that in complex structural design and 3D printing, the overhang effect caused by gravity necessitates the addition of supports during printing, resulting in additional material consumption and the need to remove supports. Thus, it achieves integrated design and manufacturing of multi-axis 3D printing with optimal self-supporting structural configuration.

[0141] During the printing process, the printing direction is dynamically adjusted to avoid the overhang effect of the structure during printing. The overall optimization design and printing of the insufficiently printed areas are achieved through angle constraint optimization, which effectively solves the problems of increased volume and significant performance degradation when printing self-supporting structures in 3-axis 3D.

Claims

1. A two-step optimized design and manufacturing method for multi-axis 3D printing, characterized in that, Includes the following steps: S1. First step design: First, the optimal structure is obtained by topology optimization without overhang constraints. Then, the printing area is divided and the local optimal printing direction is calculated for different printing areas. If the unprintable ratio is less than the threshold If the optimization terminates, proceed to step S3 if the unprintable ratio is greater than or equal to the threshold. Proceed to step S2; The optimal printing direction for each printing region is determined by the tilt direction of the boundary cells within that region. Boundary cells are cells in the adjacent set that have empty cells. The expression for determining boundary cells is... （2） In the formula, The value is used to characterize whether an element is a boundary element. When an element is a boundary element, its... The value is 1, when a certain cell is an internal cell. The value is 0; The density gradient direction of the cells is obtained by using convolution kernels, the tilt direction of the boundary cells is obtained by transformation, and the printability of the boundary cells in each printing sub-region is considered to determine the local optimal printing direction of each printing sub-region. The cell density gradient direction is obtained using a convolution kernel. The cell density gradient direction is described as follows: （3） In the formula, , , , The coordinate position of the element is defined; edge-additional elements are introduced into the design domain edge, with the density of edge-additional elements adjacent to the base set to 1, and the density of edge-additional elements in other design domains set to 0; The element density gradient direction is converted into the tilt direction of the structural boundary, which is the angle between the structural boundary and the x-axis, and is orthogonal to the element density gradient direction. For element density gradient directions in different quadrants, the corresponding structural boundary tilt directions are: （4） In the formula, For being in 0 and The direction of inclination of the structural boundary between them The unit density gradient vector; S2. Second step design: Considering the overhang constraint, the local optimal printing direction is used as the unit overhang angle constraint, and the unit horizontal neighborhood unit density constraint term is added to avoid the overhang characteristics of the optimization process. An integrated topology optimization with angular constraints is adopted, and sensitivity analysis is performed to suppress non-printable elements, thereby obtaining a self-supporting structure. The degree to which an element violates the overhang angle constraint is calculated, and the expression for the degree of violation of the overhang angle constraint is as follows: （6） In the formula, The value represents the degree of violation of the overhang angle constraint. Let be the cosine of the critical overhang angle of the structure. For unit boundary normal vector; when Tolerance If the unit satisfies the suspension angle constraint, then the unit satisfies the suspension angle constraint. The range of values ​​is ; The tilt direction of the structural boundary of each sub-region is obtained according to equation (4). Considering the overhang angle of boundary units and avoiding printing collisions between adjacent areas as constraints, the local optimal printing direction of each printing sub-region is calculated. （7） In the formula, The local optimal printing direction vector for printing sub-regions; O represents the degree vector of structural elements violating the overhang angle constraint; V is 1 for effective boundary elements and 0 for other boundary elements; M is a 0-1 mapping matrix; =( , ,…, ) represents the tilt direction vector of the structural unit; and Defined as the printing direction of two adjacent sub-regions. The maximum allowable deflection angle; when the tilt direction vector of the structural element... Within the printable range, the value of O is... Conversely, the value of O is greater than... ; The objective function is rewritten as: （8） In the formula, P is the weight of the penalty function, which is taken as 0.01; the expression for Q is: （9） In the formula, The angle between the vertical direction and the x-axis; S3, Multi-axis 3D Printing Manufacturing: Extract and optimize structural information, establish a 3D solid model, divide and slice the structure and generate a printing path to carry out multi-axis 3D printing manufacturing of self-supporting structures.

2. The two-step optimization design and manufacturing method for multi-axis 3D printing according to claim 1, characterized in that, In step S1: The method for topology optimization without overhang constraints specifically involves using a density-based SIMP model, considering free form, and determining the density of each element within the design domain involved in topology optimization. As a design variable, the expression for structural topology optimization is: （1） In the formula, This is the global displacement vector; This represents the overall nodal load vector; The total stiffness matrix; objective function The total strain energy under the action of external forces; Let i be the volume of the i-th unit. For spatial proportions; density of each unit within the design domain involving topology optimization. The value ranges from 0 to 1.

3. The two-step optimization design and manufacturing method for multi-axis 3D printing according to claim 1, characterized in that, In step S1: The printing sub-region is divided by mesh discretizing the entire design domain or by manual division method; the manual division method is to use Harris or SUSAN corner detection algorithm to obtain the structural corners of simple structures, and use the corners as vertices of the sub-region to divide the region. The resulting rectangular sub-region is the printing sub-region.

4. The two-step optimization design and manufacturing method for multi-axis 3D printing according to claim 1, characterized in that: All elements adjacent to the base are designated as invalid boundary elements, and also represent the first layer of the design domain. When a non-printable element appears in the i-th layer in the vertical printing direction, all elements below the i-th layer are considered invalid boundary elements. Elements satisfying the following formula are also considered invalid boundary elements. （5） In the formula, -θ), To determine the allowable rotation angle of the base, take... =90°, θ is the maximum overhang angle; , , , This refers to the unit coordinate position; The printing direction of invalid boundary elements was adjusted to the vertical direction after linear optimization.

5. The two-step optimization design and manufacturing method for multi-axis 3D printing according to claim 2, characterized in that, In step S2: The expression for structural topology optimization under the linear angle constraints of the cells within each printing partition is as follows: （10） In the formula, This is a constraint term for the density of cells in the horizontal neighborhood of a given cell. These are the parameter values ​​for the overhang characteristics of the structural boundary elements after topology optimization; This is a linear angle constraint term for the locally optimal printing direction in a given region. The parameters characterizing the local optimal printing direction of an element are determined by the element overhang angle. This is the global displacement vector; This represents the overall nodal load vector; The total stiffness matrix; objective function The total strain energy under the action of external forces; Let i be the volume of the i-th unit. For space ratio; For the density of each unit, The value ranges from 0 to 1.

6. The two-step optimization design and manufacturing method for multi-axis 3D printing according to claim 5, characterized in that, In step S2: the sensitivity of the objective function to the design variables is calculated according to the chain rule in order to solve the stability problem of the optimal solution; For the objective function: （11） For the constraint of the hanging angle: （12） For suspension characteristic constraints: （13） In the formula, For the Sigmoid function, The value represents the degree of violation of the overhang angle constraint. and These represent the degree to which the left and right boundaries violate the overhang angle constraint, respectively. and These represent the degree to which the left and right boundaries violate the suspension feature constraints, respectively. Let k be the displacement vector of the k-th element. The initial element stiffness matrix, The Young's modulus of the material. For parameters close to 0, As a penalty factor, take =3; Let i be the volume of the i-th unit. For the density of each unit, The value ranges from 0 to 1.

7. The two-step optimization design and manufacturing method for multi-axis 3D printing according to claim 1, characterized in that, Step S3 specifically involves: performing 3D modeling using Rhino software; slicing the solid model obtained from the 3D modeling using Cura software and generating printing paths for multi-axis 3D printing manufacturing.

Images