A self-supporting crystal structure design optimization method based on topology optimization

Abstract

This invention relates to the field of 3D printing in computer graphics, proposing a topology-optimized method for designing and optimizing self-supporting crystal structures. The technology of this invention mainly comprises two stages: subdivision and simplification. In the subdivision stage, this invention subdivides an initial coarse crystal structure consisting of several self-supporting crystal units based on a topology optimization framework. Whether a structural unit is subdivided is determined by assigning a state variable to each crystal unit. The state variables of the units in each subdivision level are converted into virtual densities of rods, allowing them to be incorporated into continuous computation. The self-support of the rods is guaranteed by a carefully designed unit topology, while the self-support at nodes (i.e., preventing node overhang) is achieved by introducing a self-supporting filter at the rod density under each newly introduced node. Repeated application of subdivision yields adaptive and optimized self-supporting crystal structures. The performance of this method has been evaluated on 2D and 3D instances, demonstrating its robustness.

Description

Technical Field

[0001] This invention belongs to the field of 3D printing in computer graphics, and relates to a design optimization method for self-supporting crystal structures based on topology optimization. Background Technology

[0002] Crystal structures possess many exciting functionalities and mechanical properties, such as negative Poisson's ratio, negative stiffness, negative compressibility, negative coefficient of thermal expansion, and extremely high stiffness and extremely low mass. Researchers have developed numerous algorithms to design crystal structures with various objectives, with typical applications in biomedical engineering and aerospace. However, the complex geometry and topology of crystal structures often make them difficult to manufacture using traditional methods. With the advent of additive manufacturing technology, the fabrication of crystal structures has become easier in recent years. However, while additive manufacturing offers design flexibility and significant geometric freedom, it also has its limitations, such as the inability to directly print overhanging regions and very thin rod-like structures. In the fabrication of crystal structures, overhanging regions naturally cannot be directly manufactured using additive manufacturing; they require additional support structures, which are difficult to remove in many cases (e.g., linear arc additive manufacturing of metals). Previous methods could not completely eliminate the need for supports. Therefore, a new approach is needed to address this issue during the structure generation stage. Summary of the Invention

[0003] This invention proposes a novel method for generating self-supporting crystal structures that optimize mechanical properties (e.g., mechanical stiffness) while maintaining desired volume constraints. The method comprises two main stages: subdivision and simplification, both formulated within a framework of topology optimization.

[0004] The technical solution of the present invention is as follows:

[0005] The technology of this invention mainly includes two stages: subdivision and simplification.

[0006] In the subdivision stage, this invention subdivides an initial coarse crystal structure composed of several self-supporting crystal units based on a topology optimization framework. Whether a structural unit is subdivided is determined by assigning a state variable to each crystal unit. The state variables of the units in each subdivision level are converted into virtual densities of rods, allowing them to be incorporated into continuous computation. The self-support of the rods is guaranteed by a carefully designed unit topology, while the self-support at the nodes (i.e., preventing node overhang) is achieved by a self-support filter introduced at the rod density under each newly introduced node. Repeated application of subdivision yields adaptive and optimized self-supporting crystal structures.

[0007] The specific implementation steps are as follows:

[0008] Step S1.1: Generate a groundlattice structure (GLS) by filling the discretized design domain with self-supporting crystal units.

[0009] Step S1.2: Parametric design domain, design unit state variables.

[0010] Step S1.3: Design filters for subdividing continuity.

[0011] Step S1.4: Transform the state variables of the crystal unit into the virtual density values ​​of the corresponding rods.

[0012] Step S1.5: Set up self-supporting filters to ensure the self-support of the rods during the subdivision process.

[0013] Step S1.6: Write out the optimization equations and perform topology optimization using the moving asymptote method.

[0014] In the simplification stage, this invention is based on a topology optimization framework, aiming to maximize mechanical stiffness. Under a given material volume, redundant rods in the self-supporting crystal structure generated in the subdivision step are further removed, while ensuring the self-support and dimensional printability of the structure.

[0015] The specific implementation steps are as follows:

[0016] Step S2.1: Establish self-supporting constraints to ensure that the structure is self-supporting during the simplification process.

[0017] Step S2.2: Design a size filter to ensure the manufacturability of the resulting structure.

[0018] Step S2.3: Write out the optimization equations and perform topology optimization using the moving asymptote method.

[0019] The beneficial effects of this invention are:

[0020] This invention designs self-supporting crystal structures based on a topology optimization framework, resulting in structures that fully utilize the material to optimize mechanical stiffness. The performance of the method has been evaluated on 2D and 3D examples, demonstrating its robustness. Attached Figure Description

[0021] Figure 1 (a) is the structure of the two-dimensional self-supporting crystal unit and the first subdivision selected in this invention;

[0022] Figure 1 (b) is the three-dimensional self-supporting crystal unit and the structure of one subdivision selected in this invention;

[0023] Figure 2(a) is a schematic diagram of the GLS generated by the two-dimensional self-supporting crystal unit of the present invention;

[0024] Figure 2 (b) is based on the present invention Figure 2 (a) shows the two-dimensional subdivision results of the boundary and load;

[0025] Figure 2 (c) is the basis of this invention Figure 2 (b) is a result of further simplification of the structure;

[0026] Figure 3 This is a parameterized schematic diagram of the segmentation strategy used in the segmentation method proposed in this invention;

[0027] Figure 4 (a) is a schematic diagram of the sub-units of the subdivided continuous filter of the present invention;

[0028] Figure 4 (b) is a schematic diagram of the parent unit corresponding to the sub-unit of the subdivided continuous filter of the present invention;

[0029] Figure 4 (c) is a schematic diagram of the adjacent units of the parent unit of the subdivided continuous filter of the present invention;

[0030] Figure 5 (a) is a schematic diagram of the self-supporting filtering method used in the subdivision method of the present invention;

[0031] Figure 5 (b) is a schematic diagram of the self-supporting crystal structure obtained by self-supporting filtration in this invention;

[0032] Figure 6 (a) is a schematic diagram of the result of simplification without considering self-supporting constraints;

[0033] Figure 6 (b) is a schematic diagram of the self-supporting constraint used in the simplified method proposed in this invention;

[0034] Figure 7 (a) is a three-dimensional bone model and its corresponding boundary and load conditions;

[0035] Figure 7 (b) is the structure obtained by subdividing the bone model using a 30% volume fraction according to the present invention;

[0036] Figure 7 (c) is the result obtained by further simplification of the present invention by using an 8% volume fraction based on the subdivision. Detailed Implementation

[0037] The present invention will now be described in further detail with reference to the accompanying drawings and two-dimensional illustrations.

[0038] The first part, the subdivision method based on the topology optimization framework, consists of the following steps:

[0039] Step S1.1: Generate GLS by filling the discretized design domain with self-supporting crystal units.

[0040] The two-dimensional and three-dimensional self-supporting crystal units selected in this invention are all self-supporting, and with each subdivision, a crystal unit will be subdivided into several identical smaller crystal units. Within a single crystal unit, each rod s can be considered as a crystal unit with a radius r. s The cylinders are all at angles not exceeding 45° to the printing direction d. The discretized design domain is then filled with self-supporting crystal units to form a GLS.

[0041] Step S1.2: Parametric design domain, design unit state variables.

[0042] For a regular 2D design domain, the resolution of GLS is expressed as: in is the resolution of the initial coarse crystal structure, and n is the maximum subdivision level of the crystal unit. During optimization, at the k-th level, for each crystal unit... Assign a state variable The subdivision of a crystal unit is used to describe whether it is subdivided, with 1 indicating subdivision and 0 indicating no subdivision. The resolution of the subdivided crystal structure in the k-th subdivision level is called the resolution of the subdivision. The following methods can be used to calculate:

[0043]

[0044] Resolution calculated using equation (1) It is the resolution of the crystal structure obtained after subdividing the (k-1) level structure. It is also a state variable The resolution.

[0045] Step S1.3: Design filters for subdividing continuity.

[0046] To ensure a smooth resolution transition in the adaptive structure, each state variable... The value for a unit It is necessary to consider its parent unit and its neighbors. The state variables of the unit. In general, the unit... state variables satisfy:

[0047]

[0048] This indicates that the subdivision states of a cell's parent cell are similar to those of its neighboring parent cells. Here (i -1 j -1 ) refers to The index of the parent unit is and in It is the floor function. (i -1 ±1,j -1 ±1) indicates that... The exponents of the cells adjacent to the parent cell. Note that a special process is required for cells located on the boundary of the design domain. To make the above requirements differentiable, a minimum p-norm approximation is used to incorporate them into the topology optimization framework. That is...

[0049]

[0050] To further push the values ​​of the state variables to 0 and 1, the state variables filtered using the Heaviside projection function are as follows:

[0051]

[0052] β controls the sharpness of the projection, while η is the threshold that controls the projection.

[0053] Step S1.4: Transform the state variables of the crystal unit into the virtual density values ​​of the corresponding rods.

[0054] For rod s, when its initial cross-sectional area is A0, ρ represents the virtual density value of s. s When optimizing, its cross-sectional area will be calculated as A. s =ρ s A0. For a rod generated by the k-th subdivision, its virtual density will be determined based on the element containing that rod. state variables To evaluate. Using the following formula:

[0055]

[0056] The state variable values ​​of a crystal unit can be mapped to the virtual density values ​​corresponding to specific bars. Where M... k It is a relation matrix that establishes the correspondence between the two, where non-zero terms are 1.

[0057] Step S1.5: Set up self-supporting filters to ensure the self-support of the rods during the subdivision process.

[0058] Although carefully designed subdivision rules can ensure self-support for all bars, node overhangs can still occur due to newly inserted nodes. The method of this invention develops a density-based filter to avoid the generation of overhang nodes. If a cell... If the element is subdivided, several rods are added to that element. For a suspended node q, this invention defines a set of rods around q, the set of which is Ω. q This includes:

[0059] 1) Connect q and the unit containing q The rod at the center node;

[0060] 2) Connect q and the unit below q. The rod at the center node.

[0061] Self-supporting filters are achieved by making Ω q The virtual density of any rod and in Ω q The maximum virtual density of all rods is the same, which is added because the maximum virtual density of all rods in the element is the same. This can be approximated by the p-norm of each bottom node q of the element:

[0062]

[0063] Where ρ s It is in Ω q The virtual density of the rod in the middle, #(Ω) q p indicates the number of elements in the set. d This represents the p-norm exponent. With the help of this self-supporting filter, the crystal structure obtained based on the TO framework will have no dangling nodes.

[0064] Step S1.6: Write the optimization equation

[0065] Based on the filters and constraints defined above, this invention provides a subdivision formula within the framework of TO. The state variable y = {y} in all levels. k The variables k = 0, 1, 2, ..., n are used together as design variables. The virtual density field ρ, transformed from the state variables and equation (4), will be used for finite element analysis and constraint calculation. This invention considers the problem of minimizing the flexibility (i.e., maximizing the stiffness) of the rod in this method, and the specific formula can be defined as:

[0066]

[0067] stK(ρ)U=F

[0068]

[0069]

[0070] The first constraint is the equilibrium equation for linear elasticity, where K, U, and F are the global stiffness matrix, nodal displacement vector, and external force vector, respectively. * Indicates the maximum available volume fraction, A0 and l s Let be the initial cross-sectional area and length of rod s. This optimization problem is solved using the gradient-based moving asymptote method.

[0071] The second part involves further simplifying the structure based on the topology optimization framework, with the following steps:

[0072] Step S2.1: Establish self-supporting constraints to ensure that the structure is self-supporting during the simplification process.

[0073] If the crystal structure is directly simplified, removing the struts below the nodes will create dangling nodes. Manufacturability is problematic for structures with dangling nodes. Therefore, a self-supporting constraint needs to be designed and imposed on all nodes during the simplification calculation.

[0074] Nodes on the bottom boundary of the design domain can always be supported by the 3D printing platform. Nodes on the top surface of the design domain will not droop, as such nodes will be automatically removed after all the rods below them are eliminated. Self-supporting constraints are only designed for internal nodes and nodes on the left / right boundaries. For node v i , its in v i The set of adjacent rods above is called In v i The adjacent rod sets below are called exist Inspect the cross-sectional area of ​​all rods and set the largest area as [value missing]. same, The largest cross-sectional area among all the rods is determined by express.

[0075] Ideally, at each node under consideration, by A structure without overhangs can be obtained. However, adding such constraints to each node leads to a large number of additional equations in the optimization framework, which are difficult to solve efficiently. Therefore, the following constraints are proposed:

[0076]

[0077] To integrate the constraints into the optimization formula, it is necessary to ensure that the constraints are differentiable. Therefore, p-norm approximation is used, and exponential transformation is employed to eliminate them. The influence of this leads to the new constraint formula:

[0078]

[0079] in and The values ​​are also approximated using the p-norm, making them differentiable. Exponential and logarithmic transformations are used to avoid the denominator of the gradient approaching zero; the formulas are as follows:

[0080]

[0081]

[0082] Step S2.2: Design a size filter to ensure the manufacturability of the resulting structure.

[0083] The goal is to calculate a radius close to zero for the redundant rods, allowing the remaining rods to be manufactured. The maximum permissible radius is r. max This can be used as a user-specified system parameter, and its value should be less than the minimum length of the GLS upper pole. To avoid merging of adjacent rods. Minimum radius r min This is determined by the minimum printable characteristics of the 3D printer. Furthermore, to ensure that each rod does not bend and eventually fail under compressive force, the radius r of each rod is... s and length l s Buckling constraints must be met:

[0084]

[0085] Where ζ is the slenderness ratio, and in summary, the radius r of the rod... s It should meet the following requirements:

[0086]

[0087] And from the cross-sectional area A of the rod s =ρ s A0 and have Therefore, the virtual density value of the rod satisfies the constraint:

[0088] ρ l ≤ρ s ≤ρ u (13)

[0089] in To remove redundant rods, those with a density less than ρ need to be removed. l The density is projected to a value close to zero. Furthermore, the projection needs to be differentiable. The projection used is as follows:

[0090]

[0091] Where λ is a parameter that controls the sharpness of the projection function.

[0092] Step S2.3: Write out the optimization equation.

[0093]

[0094] stK(ρ)U=F

[0095]

[0096] S≤1

[0097]

[0098] 0≤ρ s ≤ρ u ,s=1,2,...L

[0099] This optimization problem is also solved using the gradient-based moving asymptote method.

Claims

1. A method for designing and optimizing a self-supporting crystal structure based on topology optimization, characterized in that, The steps are as follows: Step S1: subdivision stage: Step S1.1: generate a ground lattice structure (GLS hereinafter) by filling the discretized design domain with self-supporting crystal units; Step S1.2: parameterize the design domain and design unit state variables; For regular 2D design domain, the resolution of GLS is represented as where is the resolution of the initial coarse crystal structure, n is the maximum subdivision level of the crystal unit; during optimization, for each crystal unit at the kth level a state variable is assigned to describe whether the crystal unit is subdivided, i.e. 1 for subdivided and 0 for not subdivided; the resolution of the subdivided crystal structure at the kth level is called and is calculated as follows: The resolution calculated with formula (1) is the resolution of the crystal structure obtained after subdivision of the (k-1) level structure; is also the resolution of the state variable ; Step S1.3: design a filter for the continuity of subdivision; Unit of the state variable satisfies: (i -1 , j -1 ) refers to the indices of the parent cell of ; and and where is the floor function;(i -1 ±1, j -1 ±1) denotes the indices of those cells that are adjacent to the parent cell of ; It should be noted that a special process is required for units located on the boundary of the design domain; in order to make the above requirements differentiable, a p-norm approximation of the minimum value is used to bring it into the topology optimization framework; that is The filtered state variable using the Heaviside projection function is as follows: Where β controls the sharpness of the projection, and η is the threshold value for controlling the projection; Step S1.4: convert the state variables of the crystal units into virtual density values of the corresponding rods; For rod s, when its initial cross-sectional area is A0, ρ represents the virtual density value of s. s When optimizing, its cross-sectional area will be calculated as A. s =ρ s A0; For a rod generated by the k-th subdivision, its virtual density will be based on the element containing that rod. state variables To evaluate; by the following formula: mapping the state variable values of the crystal unit to virtual density values corresponding to the specific rods; wherein M k is a relation matrix that establishes the correspondence between the two, wherein the non-zero items are 1; Step S1.5: set a self-supporting filter to ensure the self-supporting nature of the rods during the subdivision process; In the method a density-based filter is developed to avoid the creation of dangling nodes, a set of bars is defined around q, whose collection is Ω q wherein it comprises: 1) connecting q and the unit containing q the pole of the central node of 2) the link q and the unit below q the pole of the central node A self-supporting filter is added by making the virtual density of any rod in Ω q identical to the maximum virtual density of all rods in Ω q ; this can be approximated by the p-norm of each bottom node q of the cell as: where p s is the virtual density of the rods in Ω q ; #(Ω q ) indicates the number of elements in the set, p d represents the p-norm exponent; with the help of this self-supporting filter, the crystal structure obtained on the basis of the TO framework will have no dangling nodes; Step S1.6: write the optimization equation and perform topology optimization using the moving asymptote method; Step S2: simplification stage: Step S2.1: formulate self-supporting constraints to ensure that the structure is self-supporting during the simplification process; The following constraints are proposed: To integrate into the optimization formula, it is necessary to ensure that the constraint is differentiable, so the p-norm approximation is adopted, and the influence of the exponential transformation is eliminated by using the exponential transformation The new constraint formula obtained is: where and The values of p-norms are also approximated to make them differentiable; exponential and logarithmic transformations are used to avoid denominator close to zero for gradients, the formula is: Step S2.2: design a size filter to ensure the manufacturability of the resulting structure; Step S2.3: write the optimization equation and perform topology optimization using the moving asymptote method.

2. A self-supporting crystal structure design optimization method based on topology optimization according to claim 1, characterized in that, The step 1.6 is specifically operated as follows: the subdivision formula is given in the framework of TO; the state variables y = {y k ,k = 0, 1, 2,..., n} in all levels are used together as design variables; the virtual density field p converted from the state variables and formula (4) will be used for finite element analysis and constraint calculation; the minimum flexibility of the rod in this method is considered, that is, the maximum stiffness problem, and the specific formula can be defined as: min x c = U T KU (6) s.t.K(ρ)U=F where the first constraint is the linear-elastic equilibrium equation, K, U and F are the global stiffness matrix, nodal displacement vector and external force vector, respectively; V * denotes the maximum available volume fraction, A0and l s are the initial cross-sectional area and length of the bar s; the optimization problem is solved using a gradient-based moving asymptotes method.

3. A self-supporting crystal structure design optimization method based on topology optimization according to claim 1 or 2, characterized in that, In step 2.2, the specific operation is as follows: Maximum allowed radius r max Can be used as a user-specified system parameter, whose value should be smaller than the minimum length of the bars on the GLS To avoid merging of adjacent bars; minimum radius r min Is determined by the minimum printable feature of the 3D printer; furthermore, to ensure that each bar does not bend and eventually fail under compression forces, the radius r s And length l s Of each bar must satisfy the buckling constraint: where ζ is the slenderness ratio, in general, the radius r of the rod s The following should be satisfied: Again by the cross-sectional area A of the bar s = p s A0and There so that the virtual density value of the bar satisfies the constraint: p l ≤ p s ≤ p u (13) where To remove the redundant bars, it is necessary to project those with density less than p l to near zero values; moreover, the projection needs to satisfy differentiability; the projection employed is as follows: Where λ is a parameter that controls the sharpness of the projection function.

4. A self-supporting crystal structure design optimization method based on topology optimization according to claim 1 or 2, characterized in that, In step 2.3, the specific operation is as follows: Optimization equation: s.t.K(ρ)U=F The gradient-based moving asymptote method is also used to solve this optimization problem.

5. A self-supporting crystal structure design optimization method based on topology optimization according to claim 3, characterized in that, In step 2.3, the specific operation is as follows: Optimization equation: s.t.K(ρ)U=F The gradient-based moving asymptote method is also used to solve this optimization problem.

6. A self-supporting crystal structure design optimization method based on topology optimization according to claim 1 or 2 or 5, characterized in that, In step 1.1, the specific operation is as follows: The selected two-dimensional and three-dimensional self-supporting crystal units are self-supporting, and each time the subdivision is performed, one crystal unit is subdivided into several identical small crystal units; in one crystal unit, each rod s can be considered as a cylinder with a radius r s ; the included angle between them and the printing direction d is not more than 45°; then the GLS is formed by filling the discretized design domain with self-supporting crystal units.

7. A self-supporting crystal structure design optimization method based on topology optimization according to claim 3, characterized in that, In step 1.1, the specific operation is as follows: The selected two-dimensional and three-dimensional self-supporting crystal units are self-supporting, and each time the subdivision is performed, one crystal unit is subdivided into several identical small crystal units; in one crystal unit, each rod s can be considered as a cylinder with a radius r s ; the included angle between them and the printing direction d is not more than 45°; then the GLS is formed by filling the discretized design domain with self-supporting crystal units.

8. A self-supporting crystal structure design optimization method based on topology optimization according to claim 4, characterized in that, In step 1.1, the specific operation is as follows: The selected two-dimensional and three-dimensional self-supporting crystal units are self-supporting, and each time the subdivision is performed, one crystal unit is subdivided into several identical small crystal units; in one crystal unit, each rod s can be considered as a cylinder with a radius r s ; the included angle between them and the printing direction d is not more than 45°; Then fill the discretized design domain with self-supporting crystal units to form a GLS.

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